#9018149

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8925345);