#7194031

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fgi,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, 7101227);