#6756898

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6664094);