#8436229

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

4020 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,24,22)(19,29,31,25)(20,27,21,26)(23,28,32,30)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,19,24,31)(18,25,22,29)(20,30,21,28)(23,27,32,26)(34,39)(35,42)(41,44)(43,48);(17,20,24,21)(18,26,22,27)(19,30,31,28)(23,29,32,25)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,21,24,20)(18,27,22,26)(19,30,31,28)(23,29,32,25)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,22,24,18)(19,29,31,25)(20,27,21,26)(23,30,32,28)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,23,24,32)(18,28,22,30)(19,27,31,26)(20,29,21,25)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343425);