#7221144

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=hi,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,e^c=ei,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, 7128340);