#7224691

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7131887);