#8533132

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

4260 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)(3,9,11,6)(4,8,13,15)(10,16,12,14)(17,18,22,25)(19,26,21,23)(20,24,28,31)(27,32,29,30)(33,34,39,38)(35,40,45,44)(36,46,47,41)(37,42,43,48);(1,3,5,11)(2,6,7,9)(4,10,13,12)(8,14,15,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,35,39,45)(34,40,38,44)(36,37,47,43)(41,48,46,42);(2,7)(6,9)(8,15)(14,16)(20,28)(24,31)(27,29)(30,32)(33,36,39,47)(34,41,38,46)(35,43,45,37)(40,42,44,48);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37)(34,42)(35,36)(38,48)(39,43)(40,46)(41,44)(45,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,44)(43,45);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,30,31,32)(33,38,39,34)(35,44,45,40)(36,46,47,41)(37,42,43,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440328);