#7062492

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

2009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,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, 6969688);