#6575180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2275 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,21)(18,26)(20,22)(24,28)(25,27);(6,9)(10,12)(13,15,19,17)(14,21,16,23)(18,25,20,28)(22,27,26,24);(2,4)(13,16,19,14)(15,24,17,27)(18,26,20,22)(21,28,23,25);(5,7,8,11)(6,10,9,12)(13,17,19,15)(14,21,16,23)(18,28,20,25)(22,27,26,24);(13,18)(14,22)(16,26)(19,20)(21,23)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307230, #72307265, #72307285
maximal normal subgroup of	#72307230, #72307265, #72307285

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482376);