#7597082

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

2190 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=fi,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,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281581, #72282293, #72308304
maximal normal subgroup of	#72281581, #72282293, #72308304

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504278);