#135120

order	512
nilpotency class	4
isoclinism	isoclinic class of #135117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=gi,e^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=egh,f^c=fg,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42316);