#8932095

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

4115 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)(3,9,13,7)(4,8,14,5)(11,12,15,16)(17,18,23,27)(19,24,26,21)(20,29,22,25)(28,32,30,31)(33,34)(35,36)(37,40)(38,46)(39,42)(41,48)(43,47)(44,45);(1,3,6,13)(2,7,10,9)(4,11,14,15)(5,12,8,16)(17,19,23,26)(18,24,27,21)(20,30,22,28)(25,32,29,31)(33,35,39,37)(34,40,42,36)(38,43,48,45)(41,47,46,44);(4,14)(5,8)(11,15)(12,16)(17,20,23,22)(18,25,27,29)(19,28,26,30)(21,32,24,31)(33,36,39,40)(34,37,42,35)(38,47,48,44)(41,43,46,45);(1,4)(2,8)(3,11)(5,10)(6,14)(7,16)(9,12)(13,15)(17,21,23,24)(18,26,27,19)(20,31,22,32)(25,28,29,30)(33,37,39,35)(34,40,42,36)(38,45,48,43)(41,47,46,44);(1,3)(2,9)(4,15)(5,12)(6,13)(7,10)(8,16)(11,14)(17,22,23,20)(18,25,27,29)(19,28,26,30)(21,31,24,32)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(1,5,6,8)(2,4,10,14)(3,12,13,16)(7,11,9,15)(17,18)(19,24)(20,29)(21,26)(22,25)(23,27)(28,32)(30,31)(33,38,39,48)(34,41,42,46)(35,43,37,45)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839291);