#7525139

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,31)(21,25)(22,26)(24,33)(27,42)(28,47)(29,40)(30,41)(32,38)(34,46)(35,44)(36,45)(37,43)(39,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,29)(18,23,25,40)(20,32,33,27)(22,36,35,30)(24,42,31,38)(26,45,44,41)(28,37,48,34)(39,46,47,43);(10,14)(11,15)(17,20,21,33)(18,24,25,31)(19,27,29,32)(22,34,35,37)(23,38,40,42)(26,43,44,46)(28,36,48,30)(39,45,47,41);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20)(18,24)(19,28)(21,33)(22,37)(23,39)(25,31)(26,46)(27,30)(29,48)(32,36)(34,35)(38,41)(40,47)(42,45)(43,44);(2,5)(6,8)(18,25)(20,33)(23,40)(26,44)(27,32)(28,48)(34,37)(41,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7432335);