#8603533

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45467	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8510729);