#7221102

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	#25081	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(17,18,24,29)(19,25,33,43)(20,27,21,35)(22,28,23,38)(26,36,45,34)(30,40,31,46)(32,41,37,42)(39,47,44,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,24,33)(18,25,29,43)(20,31,21,30)(22,37,23,32)(26,44,45,39)(27,46,35,40)(28,42,38,41)(34,47,36,48);(10,13)(14,16)(17,20,24,21)(18,26,29,45)(19,30,33,31)(22,34,23,36)(25,39,43,44)(27,28,35,38)(32,47,37,48)(40,42,46,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,29,35)(19,31,33,30)(22,34,23,36)(25,40,43,46)(26,28,45,38)(32,47,37,48)(39,42,44,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,22,24,23)(18,28,29,38)(19,32,33,37)(20,34,21,36)(25,41,43,42)(26,35,45,27)(30,47,31,48)(39,40,44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,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,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, 7128298);