#6488326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#25754	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6395522);