#7231058

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

1156 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7138254);