#8096795

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	#45833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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:

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

Polycyclic presentation:

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

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, 8003991);