#8159132

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066328);