#7485110

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7392306);