#7298394

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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

1327 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)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,35)(27,37,38,32)(28,34,43,52)(29,36,31,33)(39,50,41,47)(40,46,44,48)(42,49,45,51)(53,56,54,55);(2,5)(3,7)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,38)(26,32,35,37)(28,41,43,39)(29,44,31,40)(33,48,36,46)(34,50,52,47)(42,53,45,54)(49,55,51,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,38)(29,31)(32,46)(34,49)(35,36)(37,48)(39,41)(42,45)(47,55)(50,56)(51,52);(2,5)(6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,39)(29,45)(30,43)(31,42)(32,47)(33,49)(35,52)(36,51)(37,50)(38,41)(40,54)(44,53)(46,55)(48,56);(1,3,4,7)(2,6,5,8)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,33,35,36)(27,40,38,44)(28,42,43,45)(32,46,37,48)(34,51,52,49)(39,53,41,54)(47,56,50,55)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,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, 7205590);