#8445676

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	#55451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,e^2=i,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, 8352872);