#8137237

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

4171 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8044433);