#7338445

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	#24244	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)(19,27)(20,31)(21,25)(22,26)(23,30)(24,28)(29,43)(32,40)(33,47)(34,38)(35,48)(36,42)(37,46)(39,44)(41,45);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,36,46)(24,38,40,31)(25,37,42,43)(33,45,35,44)(39,48,41,47);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,15,13,16)(17,20,30,32,22,34,19,28)(18,24,27,38,26,40,23,31)(21,35,46,44,36,33,29,45)(25,41,43,47,42,39,37,48);(2,6)(3,8)(9,12,14,13)(10,15,11,16)(17,21,22,36)(18,25,26,42)(19,29,30,46)(20,33,34,35)(23,37,27,43)(24,39,40,41)(28,44,32,45)(31,47,38,48);(2,6)(5,7)(9,13,14,12)(10,15,11,16)(18,26)(21,36)(23,27)(24,40)(29,46)(31,38)(33,35)(44,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,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, 7245641);