#8097000

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4194 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)(4,12)(5,9)(6,10)(7,14)(11,13)(15,16)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31)(33,34)(35,39)(36,37)(38,41)(40,45)(42,43)(44,47)(46,48);(1,3)(2,5)(4,11)(6,9)(7,13)(8,10)(12,15)(14,16)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36)(34,40)(35,42)(37,41)(38,45)(39,46)(43,47)(44,48);(1,4,6,14)(2,7,10,12)(3,11,9,16)(5,13,8,15)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,15)(12,13)(14,16)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,37,38,40)(34,36,41,45)(35,43,44,46)(39,42,47,48);(1,5,6,8)(2,9,10,3)(4,13,14,15)(7,16,12,11)(17,22,23,30)(18,25,26,32)(19,27,21,29)(20,28,24,31)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 8004196);