#10446440

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18,24,26)(19,22,23,32)(20,28,21,25)(27,31,29,30)(33,34,39,41)(35,37,43,36)(38,40,47,46)(42,45,48,44);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,24,23)(18,22,26,32)(20,29,21,27)(25,31,28,30)(33,35,39,43)(34,36,41,37)(38,42,47,48)(40,44,46,45);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20,24,21)(18,25,26,28)(19,27,23,29)(22,30,32,31)(33,36,39,37)(34,35,41,43)(38,44,47,45)(40,42,46,48);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,21,24,20)(18,25,26,28)(19,27,23,29)(22,31,32,30)(33,34)(35,36)(37,43)(38,46)(39,41)(40,47)(42,45)(44,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,22)(18,19)(20,30)(21,31)(23,26)(24,32)(25,27)(28,29)(33,37,39,36)(34,35,41,43)(38,45,47,44)(40,42,46,48);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,23,24,19)(18,22,26,32)(20,29,21,27)(25,30,28,31)(33,38,39,47)(34,40,41,46)(35,42,43,48)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 10353636);