#7613686

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2081 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=h,e^2=i,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	#72282293, #72308261
maximal normal subgroup of	#72282293, #72308261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520882);