#8926397

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

3979 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833593);