#7582306

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7489502);