#8417225

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	#41634	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(2,6)(3,9)(8,13)(11,15)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(4,12)(8,13)(11,15)(14,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324421);