#7512115

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

1544 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)(17,18,21,25)(19,27,30,23)(20,32,34,24)(22,26,36,42)(28,39,31,37)(29,38,33,40)(35,45,46,41)(43,47,44,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30)(18,23,25,27)(20,33,34,29)(22,28,36,31)(24,40,32,38)(26,37,42,39)(35,44,46,43)(41,48,45,47);(10,14)(13,15)(19,28)(22,36)(23,37)(26,42)(27,39)(29,43)(30,31)(33,44)(35,46)(38,47)(40,48)(41,45);(2,5)(6,8)(11,16)(13,15)(17,20,21,34)(18,24,25,32)(19,29,30,33)(22,35,36,46)(23,38,27,40)(26,41,42,45)(28,43,31,44)(37,47,39,48);(1,3,4,7)(2,6,5,8)(18,25)(20,34)(23,27)(26,42)(29,33)(35,46)(37,39)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,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, 7419311);