#8533319

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440515);