#9443236

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

4031 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,21,25)(19,22,27,23)(20,24,29,32)(26,30,28,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,31)(26,29)(30,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,6)(7,13)(8,11)(14,16)(17,19,21,27)(18,23,25,22)(20,26,29,28)(24,31,32,30)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,31)(26,29)(30,32)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,31)(29,32)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9350432);