#7223792

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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

1192 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)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,32,37)(27,33,42,50)(28,35,29,44)(30,36,31,46)(34,45,51,43)(38,48,39,53)(40,49,41,55)(47,54,56,52);(1,2)(3,8)(4,5)(6,7)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(25,27,32,42)(26,33,37,50)(28,38,29,39)(30,40,31,41)(34,47,51,56)(35,48,44,53)(36,49,46,55)(43,52,45,54);(2,5)(6,8)(25,28,32,29)(26,34,37,51)(27,38,42,39)(30,43,31,45)(33,47,50,56)(35,36,44,46)(40,52,41,54)(48,49,53,55);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,32,28)(26,35,37,44)(27,39,42,38)(30,43,31,45)(33,48,50,53)(34,36,51,46)(40,52,41,54)(47,49,56,55);(2,5)(6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,30,32,31)(26,36,37,46)(27,40,42,41)(28,43,29,45)(33,49,50,55)(34,44,51,35)(38,52,39,54)(47,53,56,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fhi,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=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7130988);