#6810810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#42440	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,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,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, 6718006);