#8901538

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

3899 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,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, 8808734);