#7615300

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=hi,e^2=f,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7522496);