#7615302

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(17,18,21,23)(19,27,28,24)(20,25,32,34)(22,26,33,35)(29,40,42,36)(30,41,43,37)(31,39,46,38)(44,47,45,48);(2,5)(6,7)(11,16)(13,15)(18,23)(19,28)(20,31)(22,33)(25,38)(29,44)(32,46)(34,39)(36,47)(37,41)(40,48)(42,45);(9,11,12,16)(10,13,14,15)(17,19,21,28)(18,24,23,27)(20,29,32,42)(22,30,33,43)(25,36,34,40)(26,37,35,41)(31,45,46,44)(38,48,39,47);(10,14)(13,15)(17,20,22,31,21,32,33,46)(18,25,26,39,23,34,35,38)(19,29,30,45,28,42,43,44)(24,36,37,47,27,40,41,48);(1,3,4,8)(2,6,5,7)(18,23)(19,28)(20,32)(26,35)(30,43)(31,46)(36,40)(47,48)

Polycyclic presentation:

2^9|a^2=fgi,c^2=g,d^2=h,e^2=fi,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, 7522498);