#7492753

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1212 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,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, 7399949);