#8508935

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	#35044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416131);