#8096900

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	#45584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(36,44)(40,45)(42,48)(46,47);(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(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)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29)(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)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,22,23,30)(18,25,26,32)(19,27,21,29)(20,28,24,31)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,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, 8004096);