#8691778

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	#36777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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)(4,11)(8,10)(12,13)(17,18)(19,22)(20,26)(21,31)(23,25)(24,27)(28,29)(30,32)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,38,44,43);(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,19,24,23)(18,25,27,22)(20,28,21,30)(26,32,31,29)(33,35)(34,40)(36,43)(37,42)(38,46)(39,47)(41,44)(45,48);(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,20,24,21)(18,26,27,31)(19,28,23,30)(22,29,25,32)(33,36)(34,41)(35,38)(37,40)(39,46)(42,45)(43,47)(44,48);(1,3,6,2)(4,9,11,7)(5,8,14,10)(12,15,13,16)(17,21,24,20)(18,26,27,31)(19,28,23,30)(22,32,25,29)(33,36)(34,42)(35,43)(37,40)(38,47)(39,46)(41,45)(44,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,22)(18,19)(20,29)(21,32)(23,27)(24,25)(26,28)(30,31)(33,37)(34,43)(35,42)(36,40)(38,45)(39,44)(41,47)(46,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,23,24,19)(18,25,27,22)(20,30,21,28)(26,32,31,29)(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,c^2=g,d^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, 8598974);