#7338449

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	#24455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7245645);