#10442539

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3868 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)(4,8)(5,7)(6,10)(11,16)(12,15)(13,14)(17,18,23,26)(19,24,22,20)(21,29,31,25)(27,28,30,32)(33,34,39,37)(35,40,46,45)(36,42,47,41)(38,48,44,43);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,23,22)(18,24,26,20)(21,30,31,27)(25,28,29,32)(33,35,39,46)(34,40,37,45)(36,44,47,38)(41,48,42,43);(1,3,7,4)(2,9,5,8)(6,11,14,12)(10,16,13,15)(17,20)(18,22)(19,26)(21,28)(23,24)(25,30)(27,29)(31,32)(33,36)(34,41)(35,44)(37,42)(38,46)(39,47)(40,48)(43,45);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20)(18,19)(21,28)(22,26)(23,24)(25,27)(29,30)(31,32)(33,37,39,34)(35,45,46,40)(36,42,47,41)(38,48,44,43);(1,5,7,2)(3,8,4,9)(6,13,14,10)(11,15,12,16)(17,21,23,31)(18,25,26,29)(19,27,22,30)(20,28,24,32)(33,36)(34,42)(35,44)(37,41)(38,46)(39,47)(40,43)(45,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22,23,19)(18,24,26,20)(21,30,31,27)(25,32,29,28)(33,38,39,44)(34,43,37,48)(35,36,46,47)(40,41,45,42)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10349735);