#8630805

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	#55915	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,24,28)(19,22,23,25)(20,27,21,26)(29,30,31,32)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,3,7,11)(2,8,5,15)(4,12,14,10)(6,9,16,13)(17,19,24,23)(18,25,28,22)(20,29,21,31)(26,30,27,32)(34,36)(35,41)(40,44)(43,47);(2,5)(3,11)(9,13)(10,12)(17,20,24,21)(18,26,28,27)(19,29,23,31)(22,32,25,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,7,14)(2,9,5,13)(3,10,11,12)(6,15,16,8)(17,21,24,20)(18,27,28,26)(19,29,23,31)(22,32,25,30)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,5,7,2)(3,8,11,15)(4,13,14,9)(6,12,16,10)(17,22,24,25)(18,19,28,23)(20,30,21,32)(26,31,27,29)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,6)(2,10)(3,9)(4,8)(5,12)(7,16)(11,13)(14,15)(17,23,24,19)(18,25,28,22)(20,31,21,29)(26,30,27,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=h,e^2=gi,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, 8538001);