#6636867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636852
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,19)(20,28,30,32)(21,25,22,27)(24,31,26,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,18)(20,24,30,26)(21,25,22,27)(28,31,32,29);(11,15)(13,16)(17,20,21,31,23,30,22,29)(18,24,27,32,19,26,25,28);(2,6)(3,8)(11,15)(13,16)(17,21,23,22)(18,25,19,27)(20,29,30,31)(24,32,26,28);(2,6)(5,7)(10,14)(13,16)(18,19)(24,26)(25,27)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di

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, 6544063);