#7194028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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)(27,32,30,45)(28,38,41,48)(29,34,44,52)(33,36,49,40)(37,46,43,51)(39,55,42,56)(47,53,50,54);(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)(26,32,35,45)(28,36,41,40)(29,37,44,43)(33,38,49,48)(34,46,52,51)(39,53,42,54)(47,55,50,56);(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)(26,33,32,38,35,49,45,48)(29,42,43,53,44,39,37,54)(34,50,46,56,52,47,51,55);(1,4,5,3)(2,7,6,8)(9,13,15,14)(10,17,12,22)(11,19,21,20)(16,23,18,24)(25,29,31,44)(26,34,35,52)(27,37,30,43)(28,39,41,42)(32,46,45,51)(33,47,49,50)(36,53,40,54)(38,55,48,56);(9,14,15,13)(10,17,12,22)(11,20,21,19)(16,23,18,24)(25,27,31,30)(26,32,35,45)(28,40,41,36)(29,43,44,37)(33,48,49,38)(34,51,52,46)(39,53,42,54)(47,55,50,56)

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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7101224);