#6749050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(3,5,7,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,30,33)(27,35,38,48)(28,32,39,46)(29,41,43,34)(31,42,45,37)(36,49,50,56)(40,53,52,47)(44,54,55,51);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(28,39)(32,46)(36,50)(40,52)(44,55)(47,53)(49,56)(51,54);(3,7)(5,8)(25,27,30,38)(26,31,33,45)(28,36,39,50)(29,42,43,37)(32,44,46,55)(34,48,41,35)(40,54,52,51)(47,56,53,49);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,30,39)(26,32,33,46)(27,36,38,50)(29,40,43,52)(31,44,45,55)(34,47,41,53)(35,49,48,56)(37,51,42,54);(3,7)(5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(26,33)(28,39)(31,45)(34,41)(35,48)(36,50)(40,52)(51,54)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6656246);