#6816468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39963	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=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, 6723664);