#8806838

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

4213 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,34,38,39)(35,40,43,37)(36,41,46,47)(42,48,44,45);(2,7)(4,12)(5,8)(11,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(34,39)(37,40)(41,47)(45,48);(2,7)(3,10)(4,12)(13,15)(18,22)(19,26)(24,28)(27,32)(33,35,38,43)(34,40,39,37)(36,44,46,42)(41,45,47,48);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,36)(34,41)(35,42)(37,45)(38,46)(39,47)(40,48)(43,44);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,35)(36,45)(38,40)(39,43)(41,42)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714034);