#8137633

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

4253 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)(3,10)(4,12)(14,15)(17,18)(19,28)(20,25)(21,31)(22,32)(23,27)(24,30)(26,29)(35,42)(37,45)(39,40)(41,44);(1,2)(3,7)(4,9)(5,6)(8,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,22)(21,29)(23,30)(25,26)(27,28)(31,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(1,3)(2,7)(4,11)(5,10)(6,8)(9,14)(12,16)(13,15)(17,20,23,21)(18,25,27,31)(19,22,30,29)(24,26,28,32)(33,35)(34,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48);(1,3)(2,8)(4,11)(5,10)(6,7)(9,15)(12,16)(13,14)(17,21,23,20)(18,25,27,31)(19,29,30,22)(24,26,28,32)(33,36,38,34)(35,40,42,39)(37,44,45,41)(43,47,48,46);(1,2)(3,7)(4,9)(5,6)(8,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,31)(22,26)(23,27)(28,30)(29,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,5,12)(2,9,6,13)(3,11,10,16)(7,14,8,15)(17,22,23,29)(18,26,27,32)(19,20,30,21)(24,25,28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,f^2=g,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, 8044829);