#8696018

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	#49753	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4199 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603214);