#7596782

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2203 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,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, 7503978);