#7609941

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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,32,41)(20,25,30,39)(21,26,36,45)(22,27,37,46)(29,38,47,48)(31,40,33,42)(34,43,35,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,28,41)(20,29,30,47)(21,35,36,34)(22,33,37,31)(25,38,39,48)(26,44,45,43)(27,42,46,40);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,30)(24,38)(26,42)(27,43)(28,39)(31,36)(32,47)(35,37)(40,45)(41,48)(44,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,29,23,36,37,47)(18,26,27,38,28,45,46,48)(19,30,31,35,32,20,33,34)(24,39,40,44,41,25,42,43);(2,6)(7,8)(20,30)(21,36)(25,39)(26,45)(29,47)(34,35)(38,48)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,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, 7517137);