#8097001

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

4198 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:

(2,7)(3,9)(10,15)(11,16)(17,18,23,27)(19,21,22,24)(20,26,31,25)(28,32,29,30)(33,34)(35,39)(36,37)(38,41)(40,45)(42,43)(44,47)(46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,19,23,22)(18,24,27,21)(20,29,31,28)(25,30,26,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,32)(26,27)(33,36)(34,40)(35,42)(37,41)(38,45)(39,46)(43,47)(44,48);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,21,23,24)(18,19,27,22)(20,30,31,32)(25,28,26,29)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,20)(18,26)(19,29)(21,30)(22,28)(23,31)(24,32)(25,27)(33,37,38,40)(34,36,41,45)(35,43,44,46)(39,42,47,48);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,22,23,19)(18,24,27,21)(20,29,31,28)(25,32,26,30)(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=gh,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, 8004197);