#6465857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 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)(7,10)(8,9)(11,12)(14,18)(15,21)(17,24)(19,26)(22,27)(23,28);(2,3)(6,9)(10,12)(13,14)(15,22)(16,18)(17,20)(19,23)(21,26)(24,25)(27,28);(1,2,4,3)(13,15,17,21,16,23,24,28)(14,19,20,27,18,22,25,26);(1,3,4,2);(5,7,8,11)(6,10,9,12)(14,18)(19,22)(20,25)(26,27)

Polycyclic presentation:

2^9|c^2=ghi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285116, #72285267
maximal normal subgroup of	#72285116, #72285267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373053);