#7529281

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1462 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7436477);