#8691782

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	#36339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4121 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)(7,14)(8,11)(15,16)(18,20)(19,25)(24,28)(27,29)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,38,44,43);(1,2,5,6)(3,7,10,14)(4,8,13,11)(9,15,12,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,35)(34,40)(36,43)(37,42)(38,46)(39,47)(41,44)(45,48);(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36)(34,41)(35,38)(37,40)(39,46)(42,45)(43,47)(44,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,36)(34,42)(35,43)(37,40)(38,47)(39,46)(41,45)(44,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,20,22,18)(19,26,25,23)(21,24,30,28)(27,31,29,32)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,43)(34,44,45,37)(35,46,47,36)(40,41,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8598978);