#9913253

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820449);