#7223806

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

1187 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)(27,33)(28,41)(29,35)(30,36)(31,46)(32,37)(34,43)(38,48)(39,49)(40,51)(42,56)(44,53)(45,50)(47,55)(52,54);(1,2)(3,8)(4,5)(6,7)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(25,27,32,40)(26,33,37,51)(28,42,29,38)(30,45,31,39)(34,52,53,47)(35,48,41,56)(36,50,46,49)(43,54,44,55);(2,5)(6,8)(25,28,32,29)(26,34,37,53)(27,38,40,42)(30,43,31,44)(33,47,51,52)(35,46,41,36)(39,54,45,55)(48,50,56,49);(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)(26,35,37,41)(27,38,40,42)(30,43,31,44)(33,48,51,56)(34,46,53,36)(39,55,45,54)(47,50,52,49);(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)(27,39,40,45)(28,43,29,44)(33,49,51,50)(34,35,53,41)(38,54,42,55)(47,48,52,56)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,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, 7131002);