#8115517

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(18,20)(21,29)(23,26)(27,31)(33,34,39,37)(35,40,45,46)(36,42,38,41)(43,48,44,47);(2,6)(7,9)(8,12)(14,15)(18,20)(19,25)(24,28)(27,31)(33,35)(34,40)(36,43)(37,46)(38,44)(39,45)(41,47)(42,48);(3,10)(7,9)(11,16)(14,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,36,39,38)(34,41,37,42)(35,43,45,44)(40,47,46,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,37,39,34)(35,40,45,46)(36,42,38,41)(43,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,38,39,36)(34,42,37,41)(35,44,45,43)(40,48,46,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)

Polycyclic presentation:

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

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, 8022713);