#149639

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11448
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2905 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

243 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=efhi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56835);