#6993870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(9,10,15,12)(11,18,20,16)(13,21,14,17)(19,23,22,24)(25,26)(27,37)(28,41)(29,36)(30,32)(31,35)(33,40)(34,42)(38,50)(39,53)(43,48)(44,49)(45,46)(47,54)(51,52)(55,56);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,23)(21,24)(26,32)(27,38)(28,42)(35,36)(40,45)(43,44)(47,56)(51,53);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(25,27)(26,33)(28,39)(29,40)(30,38)(31,45)(32,46)(34,47)(35,37)(36,50)(41,56)(42,52)(43,54)(44,55)(48,53)(49,51);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28,30,42)(26,34,32,41)(27,39,38,52)(29,44,31,43)(33,47,46,56)(35,49,36,48)(37,51,50,53)(40,55,45,54);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(25,29,30,31)(26,35,32,36)(27,40,38,45)(28,43,42,44)(33,37,46,50)(34,48,41,49)(39,54,52,55)(47,53,56,51)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fh,d^2=fhi,e^2=fhi,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,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6901066);