#8345932

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	#59860	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8253128);