#8118853

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	#37048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026049);