#7608587

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7515783);