#6506751

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(13,14,19,23)(15,24,28,39)(16,21,29,35)(17,22,18,32)(20,26,34,27)(25,38,40,44)(30,36,31,37)(33,42,43,41);(5,6)(7,11)(8,10)(9,12)(13,15,18,27,19,28,17,26)(14,20,22,24,23,34,32,39)(16,25,30,41,29,40,31,42)(21,33,37,38,35,43,36,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(16,29)(21,35)(25,40)(30,31)(33,43)(36,37)(38,44)(41,42);(7,12)(9,11)(13,16)(14,21)(15,25)(17,31)(18,30)(19,29)(20,33)(22,37)(23,35)(24,38)(26,42)(27,41)(28,40)(32,36)(34,43)(39,44);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,30,29,31)(20,24,34,39)(21,36,35,37)(25,41,40,42)(33,44,43,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413947);