#6676693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

2025 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)(8,12)(9,11)(13,14)(15,21)(16,19)(17,20)(18,24)(22,25)(23,28)(26,27);(5,7,10,6)(8,12,9,11)(13,15,16,24)(14,18,19,21)(17,23,26,25)(20,22,27,28);(2,4)(13,14)(15,22)(16,19)(17,27)(18,23)(20,26)(21,25)(24,28);(5,8,10,9)(6,11,7,12)(14,19)(18,21)(20,27)(22,28);(5,9,10,8)(6,11,7,12)(15,23)(17,26)(18,22)(20,27)(21,28)(24,25)

Polycyclic presentation:

2^9|b^2=f,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308154, #72308293
maximal normal subgroup of	#72308154, #72308293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583889);