#6578337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2284 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)(5,6,9,10)(7,11,8,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,28)(25,27);(6,10)(11,12)(13,15,17,26)(14,19,21,23)(16,24,25,18)(20,22,27,28);(2,4)(15,16)(18,24)(19,27)(20,23)(22,28)(25,26);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,25)(18,26)(19,22)(21,27)(23,28);(5,8,9,7)(6,12,10,11)(15,25)(16,26)(18,24)(19,20)(22,28)(23,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,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	#72307099, #72307139, #72307307
maximal normal subgroup of	#72307099, #72307139, #72307307

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485533);