#6459565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2599 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308307, #72308312, #72308349
maximal normal subgroup of	#72308307, #72308312, #72308349

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366761);