#6891072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1339 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)(11,19)(12,16)(13,18)(14,17)(15,22)(20,23)(21,24)(25,26,30,35)(27,37,39,32)(28,34,29,42)(31,46,45,36)(33,44,50,41)(38,48,43,51)(40,49,55,53)(47,54,52,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,19)(12,20,18,24)(13,23,16,21)(25,27,30,39)(26,32,35,37)(28,38,29,43)(31,40,45,55)(33,47,50,52)(34,51,42,48)(36,49,46,53)(41,54,44,56);(1,3)(2,5)(4,8)(6,7)(9,12,14,18)(10,16,17,13)(11,20,22,24)(15,21,19,23)(25,28)(26,33)(27,38)(29,30)(31,44)(32,47)(34,46)(35,50)(36,42)(37,52)(39,43)(40,56)(41,45)(48,49)(51,53)(54,55);(3,8)(5,7)(10,17)(12,18)(15,19)(20,24)(25,29,30,28)(26,34,35,42)(27,38,39,43)(31,44,45,41)(32,48,37,51)(33,36,50,46)(40,54,55,56)(47,53,52,49);(2,6)(5,7)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(27,39)(28,29)(32,37)(33,50)(34,42)(40,55)(41,44)(49,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,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, 6798268);