#10534340

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	#48136	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

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, 10441536);