#6685542

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308009, #72308034
maximal normal subgroup of	#72308009, #72308034

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592738);