#7248065

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(11,15)(12,16)(13,14)(17,18,24,28,23,29,22,39)(19,30,35,46,34,25,33,42)(20,27,36,43,21,37,38,26)(31,45,47,40,32,41,48,44);(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(17,19,23,34)(18,25,29,30)(20,31,21,32)(22,33,24,35)(26,40,43,44)(27,41,37,45)(28,42,39,46)(36,47,38,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,15)(13,14)(17,20,23,21)(18,26,29,43)(19,31,34,32)(22,36,24,38)(25,40,30,44)(27,28,37,39)(33,47,35,48)(41,42,45,46);(3,8)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,27,29,37)(19,32,34,31)(22,36,24,38)(25,41,30,45)(26,28,43,39)(33,47,35,48)(40,42,44,46);(2,6)(5,7)(9,12,13,11)(10,15,14,16)(17,22,23,24)(18,28,29,39)(19,33,34,35)(20,36,21,38)(25,42,30,46)(26,37,43,27)(31,47,32,48)(40,45,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,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, 7155261);