#6798174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#37775	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6705370);