#8621816

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
prime radical	#40846	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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)(7,9)(8,12)(14,15)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(34,40)(35,36)(42,47)(43,44);(3,10)(7,9)(11,16)(14,15)(19,24)(21,29)(27,30)(28,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=hi,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, 8529012);