#8926417

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	#47079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(17,18,23,24)(19,25,20,26)(21,27,32,22)(28,29,30,31)(33,34,39,37)(35,36,42,40)(38,45,46,41)(43,47,48,44);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(18,24)(21,32)(25,26)(28,30)(34,37)(35,42)(41,45)(43,48);(2,7)(5,12)(8,15)(11,16)(17,19,23,20)(18,25,24,26)(21,28,32,30)(22,31,27,29)(33,35,39,42)(34,40,37,36)(38,43,46,48)(41,47,45,44);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,20,23,19)(18,26,24,25)(21,28,32,30)(22,31,27,29)(33,36)(34,35)(37,42)(38,44)(39,40)(41,43)(45,48)(46,47);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,21)(18,27)(19,28)(20,30)(22,24)(23,32)(25,29)(26,31)(33,37,39,34)(35,36,42,40)(38,45,46,41)(43,44,48,47);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,22,23,27)(18,21,24,32)(19,29,20,31)(25,30,26,28)(33,38,39,46)(34,41,37,45)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=gh,e^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, 8833613);