#6506717

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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

3108 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)(15,23)(16,21)(17,29)(18,20)(19,25)(22,30)(24,36)(26,35)(27,33)(28,41)(31,39)(32,38)(34,40)(37,43)(42,44);(5,6)(7,11)(8,10)(9,12)(13,15,17,25,18,26,30,39)(14,19,22,23,20,31,29,35)(16,24,28,38,27,37,40,44)(21,32,34,36,33,42,41,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,20)(19,31)(21,33)(22,29)(23,35)(32,42)(34,41)(36,43);(7,12)(9,11)(13,16,18,27)(14,21,20,33)(15,24,26,37)(17,28,30,40)(19,32,31,42)(22,34,29,41)(23,36,35,43)(25,38,39,44);(6,10)(9,11)(16,27)(21,33)(24,37)(28,40)(32,42)(34,41)(36,43)(38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413913);