#6816475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42928	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 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)(15,17)(16,18)(19,22)(20,21)(23,24)(25,28)(26,27);(5,7,8,12)(6,9,10,11)(13,15,19,21)(14,20,22,17)(16,18,26,27)(23,24,28,25);(1,2,3,4)(6,10)(7,12)(13,16,20,25,19,26,17,24)(14,18,21,28,22,27,15,23);(6,10)(9,11)(13,17,19,20)(14,21,22,15)(16,25,26,24)(18,23,27,28);(13,18,17,23,19,27,20,28)(14,16,15,24,22,26,21,25)

Polycyclic presentation:

2^9|b^2=f,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,d^c=di,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, 6723671);