#8508919

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

4018 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,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, 8416115);