#6684847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682420
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2289 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282960, #72283292, #72285246
maximal normal subgroup of	#72282960, #72283292, #72285246

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592043);