#7194009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7101205);