#7224584

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

1169 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)(27,32)(28,40)(29,35)(30,46)(31,36)(33,42)(34,43)(37,54)(38,49)(39,51)(41,48)(44,50)(45,53)(47,56)(52,55);(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)(28,41)(29,44)(30,38)(31,39)(33,52)(34,54)(35,50)(36,51)(37,43)(40,48)(42,55)(45,56)(46,49)(47,53);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28)(26,33)(27,37)(29,42)(30,45)(31,43)(32,47)(34,35)(36,53)(38,55)(39,41)(40,46)(44,56)(48,50)(49,54)(51,52);(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)(27,37)(29,45)(30,42)(31,43)(32,48)(33,35)(36,40)(38,56)(39,41)(44,55)(46,53)(47,50)(49,52)(51,54);(2,5)(6,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,30)(26,35,36,46)(27,38,39,44)(28,42,43,45)(32,49,51,50)(33,34,53,40)(37,55,41,56)(47,54,52,48)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 7131780);