#6458528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35272	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2174 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

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, 6365724);