#7330691

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7237887);