#498755

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23350
prime radical	#575	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,15,9,16,38,56,7,14,36,54,6,13,35,53)(3,17,21,58,20,42,61,44,22,51,24,48,33,10,29,41)(4,23,19,60,25,52,28,62,26,11,43,40,34,45,63,47)(5,27,18,57,31,50,59,55,32,49,37,64,30,12,46,39);(1,3)(2,10)(4,18)(5,28)(6,33)(7,22)(8,24)(9,20)(11,39)(12,47)(13,51)(14,42)(15,44)(16,17)(19,30)(21,36)(23,55)(25,59)(26,37)(27,60)(29,38)(31,43)(32,63)(34,46)(35,61)(40,49)(41,54)(45,57)(48,56)(50,62)(52,64)(53,58);(1,4,7,26)(2,11,14,23)(3,18)(5,29)(6,34,9,25)(8,19,36,43)(10,39)(12,48)(13,52,16,45)(15,40,54,60)(17,57)(20,59)(21,31)(22,37)(24,30)(27,41)(28,35,63,38)(32,61)(33,46)(42,55)(44,49)(47,53,62,56)(50,58)(51,64);(1,5,6,30,7,32,9,31)(2,12,13,49,14,50,16,27)(3,19,20,28,22,43,33,63)(4,24,25,29,26,21,34,61)(8,37,35,59,36,18,38,46)(10,40,17,47,42,60,51,62)(11,44,45,48,23,41,52,58)(15,55,53,57,54,39,56,64)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cefh,d^b=de,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405951);