#8745963

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653159);