#7221101

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 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)(17,18,23,29)(19,31,34,25)(20,27,21,37)(22,30,24,28)(26,38,44,36)(32,41,35,46)(33,42,39,45)(40,48,43,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,23,34)(18,25,29,31)(20,35,21,32)(22,39,24,33)(26,43,44,40)(27,41,37,46)(28,45,30,42)(36,47,38,48);(10,13)(14,16)(17,20,23,21)(18,26,29,44)(19,32,34,35)(22,36,24,38)(25,40,31,43)(27,30,37,28)(33,47,39,48)(41,45,46,42);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,37)(19,32,34,35)(22,36,24,38)(25,41,31,46)(26,30,44,28)(33,48,39,47)(40,45,43,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,22,23,24)(18,28,29,30)(19,33,34,39)(20,36,21,38)(25,42,31,45)(26,27,44,37)(32,47,35,48)(40,41,43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=f,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, 7128297);