#8139469

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

3907 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046665);