#7613852

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

1980 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);(2,3)(6,9)(10,11)(13,14,19,22)(15,20,25,28)(16,17,21,18)(23,24,26,27);(1,2,4,3)(13,15,19,25)(14,20,22,28)(16,26,21,23)(17,27,18,24);(1,3,4,2)(13,16,18,14,19,21,17,22)(15,23,24,28,25,26,27,20);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,21,22,16)(15,24,25,27)(20,26,28,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281922, #72282293, #72308452
maximal normal subgroup of	#72281922, #72282293, #72308452

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521048);