#8926253

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

3993 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:

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

Polycyclic presentation:

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

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, 8833449);