#8926663

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	#37055	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,18,23,21)(19,20,26,24)(22,29,30,25)(27,31,32,28)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(18,21)(19,26)(25,29)(27,32)(37,45)(41,44)(43,46)(47,48);(1,3,6,4)(2,7,9,10)(5,12,14,11)(8,16,13,15)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38,39,35)(34,42,36,40)(37,46,45,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833859);