#7062442

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	#44913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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)(7,9)(8,10)(11,12)(13,14,19,22,18,16,15,20)(17,24,25,27,26,21,23,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,18,19)(14,20,16,22)(17,23,26,25)(21,27,24,28);(5,7,8,12)(6,9,10,11)(13,16,18,14)(15,20,19,22)(17,21,26,24)(23,28,25,27);(2,4)(7,12)(9,11)(13,17,18,26)(14,21,16,24)(15,23,19,25)(20,27,22,28);(6,10)(9,11)(13,15,18,19)(14,22,16,20)(17,25,26,23)(21,27,24,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

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, 6969638);