#7288289

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7195485);