#7298321

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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)(27,36)(28,33)(29,34)(30,32)(31,37)(35,43)(38,48)(39,44)(40,45)(41,49)(42,47)(46,51)(50,54)(52,55)(53,56);(2,5)(3,7)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,40)(29,37)(30,39)(32,46)(33,48)(34,36)(35,44)(38,41)(42,53)(43,51)(45,49)(47,55)(50,56)(52,54);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,37)(30,43)(31,44)(33,47)(34,35)(36,46)(38,40)(42,54)(45,55)(48,56)(49,50);(2,5)(6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,41)(26,33,34,49)(27,38,37,40)(30,42,43,54)(31,45,36,48)(32,47,35,50)(39,52,51,53)(44,55,46,56);(1,3,4,7)(2,6,5,8)(10,18)(15,19)(16,17)(23,24)(26,34)(27,37)(28,41)(32,35)(39,51)(42,54)(45,48)(55,56)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=f,e^2=hi,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,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, 7205517);