#7062885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#23962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1893 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,20,23,22)(17,19,25,21)(24,26,28,27);(5,6)(7,11)(8,10)(9,12)(14,19)(16,21)(17,25)(20,26)(22,27)(24,28);(2,4)(5,7,8,12)(6,9,10,11)(13,15,18,23)(14,20,16,22)(17,24,25,28)(19,26,21,27);(7,12)(9,11)(13,16,18,14)(15,22,23,20)(17,21,25,19)(24,27,28,26);(6,10)(9,11)(15,23)(20,22)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6970081);