#7614939

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(19,27)(20,31)(23,28)(24,38)(30,41)(32,46)(33,39)(37,43);(2,6)(7,8)(11,16)(13,15)(17,18,21,25)(19,28,27,23)(20,32,31,46)(22,35,34,26)(24,39,38,33)(29,44,40,47)(30,37,41,43)(36,45,42,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,29)(21,27)(22,30)(24,36)(25,28)(26,37)(31,40)(32,47)(33,45)(34,41)(35,43)(38,42)(39,48)(44,46);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,22,33,21,31,34,39)(18,24,26,32,25,38,35,46)(19,29,30,45,27,40,41,48)(23,36,37,47,28,42,43,44);(2,6)(7,8)(19,27)(20,31)(23,28)(24,38)(30,41)(32,46)(33,39)(37,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7522135);