#8587971

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

4164 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=g,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, 8495167);