#9494729

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

4325 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^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,e^c=egi,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, 9401925);