#6576488

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2859 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)(5,6)(7,12)(8,10)(9,11)(13,14,18,21)(15,16,22,20)(17,23,19,24)(25,28,26,27);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,21)(17,26)(18,22)(19,25)(23,28)(24,27);(2,4)(14,21)(15,25)(16,27)(17,19)(20,28)(22,26);(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,21)(17,27)(18,20)(19,28)(23,25)(24,26);(13,17)(14,23)(18,19)(21,24)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263962, #72263990, #72264055
maximal normal subgroup of	#72263962, #72263990, #72264055
maximal quotient of	#72265094, #72308309

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483684);