#7047334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,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, 6954530);