#8096975

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

4233 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)(3,7,4,8)(5,9,13,16)(11,14,12,15)(17,18)(19,23)(20,21)(22,25)(24,29)(26,27)(28,31)(30,32)(33,34)(35,42)(36,45)(37,44)(38,40)(39,41)(43,47)(46,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,44)(34,37,41,42)(36,46,38,43)(40,48,45,47);(1,3,6,4)(2,7,10,8)(5,11,13,12)(9,14,16,15)(17,20)(18,24)(19,26)(21,25)(22,29)(23,30)(27,31)(28,32)(33,36,39,38)(34,40,41,45)(35,43,44,46)(37,47,42,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,35)(36,47)(38,48)(39,42)(40,43)(41,44)(45,46);(1,4,6,3)(2,8,10,7)(5,11,13,12)(9,14,16,15)(17,21,22,24)(18,20,25,29)(19,27,28,30)(23,26,31,32)(33,34)(35,37)(36,40)(38,45)(39,41)(42,44)(43,47)(46,48);(1,5,6,13)(2,9,10,16)(3,11,4,12)(7,14,8,15)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,38,39,36)(34,40,41,45)(35,43,44,46)(37,48,42,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^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, 8004171);