#6481998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,31)(27,35,37,48)(28,33,39,45)(30,42,41,34)(32,43,44,38)(36,49,50,56)(40,53,52,47)(46,54,55,51);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(26,31)(28,39)(32,44)(34,42)(35,48)(36,50)(40,52)(51,54);(11,17)(14,21)(18,19)(22,23)(25,27,29,37)(26,32,31,44)(28,36,39,50)(30,43,41,38)(33,46,45,55)(34,48,42,35)(40,54,52,51)(47,56,53,49);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28,29,39)(26,33,31,45)(27,36,37,50)(30,40,41,52)(32,46,44,55)(34,47,42,53)(35,49,48,56)(38,51,43,54);(10,16)(14,21)(15,24)(22,23)(28,39)(33,45)(36,50)(40,52)(46,55)(47,53)(49,56)(51,54)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi,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, 6389194);