#7298384

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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)(27,37)(28,34)(29,36)(30,35)(31,33)(32,38)(39,50)(40,46)(41,47)(42,49)(43,52)(44,48)(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)(26,32)(28,41)(29,44)(30,38)(31,40)(33,48)(34,50)(35,37)(36,46)(39,43)(42,53)(45,54)(47,52)(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)(25,28,30,43)(26,34,35,52)(27,39,38,41)(29,45,31,42)(32,47,37,50)(33,49,36,51)(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|b^2=fi,c^2=fi,d^2=f,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, 7205580);