#7512182

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7419378);