#8741897

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

4140 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8649093);