#7007188

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2077 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)(13,14,20,15,19,17,18,22)(16,21,25,23,27,24,26,28);(5,6)(7,11)(8,10)(9,12)(13,15,19,22)(14,18,17,20)(16,23,27,28)(21,26,24,25);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,27)(14,21,17,24)(15,23,22,28)(18,26,20,25);(7,12)(9,11)(13,17,18,15,19,14,20,22)(16,24,26,23,27,21,25,28);(6,10)(9,11)(13,18,19,20)(14,15,17,22)(16,25,27,26)(21,28,24,23)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6914384);