#10352440

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	#57669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 10259636);