#6545216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2177 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257681, #72264121, #72264317
maximal normal subgroup of	#72257681, #72264121, #72264317
maximal quotient of	#72308534

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452412);