#9243244

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

4164 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,34,38,41)(35,37,44,47)(36,40,46,39)(42,48,43,45);(3,9)(6,13)(10,15)(14,16)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,35,38,44)(34,37,41,47)(36,43,46,42)(39,48,40,45);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,36,38,46)(34,39,41,40)(35,42,44,43)(37,45,47,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,43)(38,44)(39,48)(40,45)(41,47)(42,46);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37)(34,35)(36,45)(38,47)(39,42)(40,43)(41,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9150440);