#7328778

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34)(27,35,36,50)(28,32,40,47)(29,43,37,33)(31,44,45,39)(38,51,41,54)(42,56,52,49)(46,55,48,53);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,36)(28,40)(31,45)(32,47)(35,50)(39,44)(42,52)(49,56);(1,3)(2,5)(4,8)(6,7)(25,27,37,44,30,36,29,39)(26,31,43,50,34,45,33,35)(28,41,52,53,40,38,42,55)(32,48,56,51,47,46,49,54);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,40)(26,32,34,47)(27,38,36,41)(29,42,37,52)(31,46,45,48)(33,49,43,56)(35,51,50,54)(39,53,44,55);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(27,36)(28,40)(31,45)(32,47)(35,50)(39,44)(42,52)(49,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7235974);