#6558186

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2663 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)(13,14)(15,24)(16,21)(17,22)(18,28)(19,23)(20,26)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,21,19,25,18,26)(14,20,28,27,23,16,22,24);(1,2,3,4)(6,9)(10,12)(15,25)(16,20)(21,26)(24,27);(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,22)(17,24)(18,27)(19,20)(23,26)(25,28);(7,11)(10,12)(13,17,19,18)(14,22,23,28)(15,21,25,26)(16,27,20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264121, #72264159, #72264241
maximal normal subgroup of	#72264121, #72264159, #72264241
maximal quotient of	#72264552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465382);