#8587167

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

4188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494363);