#6993841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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

1345 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)(27,35)(28,39)(29,31)(30,43)(32,38)(33,40)(34,42)(36,48)(37,51)(41,47)(44,45)(46,56)(49,50)(52,53)(54,55);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,23)(21,24)(26,31)(27,36)(33,39)(34,43)(37,50)(38,44)(47,54)(52,56);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(25,27,29,36)(26,32,31,45)(28,37,40,50)(30,44,42,38)(33,46,39,53)(34,48,43,35)(41,56,55,52)(47,51,54,49);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28,29,40)(26,33,31,39)(27,37,36,50)(30,41,42,55)(32,46,45,53)(34,47,43,54)(35,49,48,51)(38,52,44,56);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(28,40)(33,39)(37,50)(41,55)(46,53)(47,54)(49,51)(52,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6901037);