#7537910

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1982 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=g,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7445106);