#7193986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7101182);