#8751950

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

4175 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:

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

Polycyclic presentation:

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

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, 8659146);