#8437816

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(20,24)(21,25)(22,26)(23,27)(29,30)(31,32)(34,39)(35,44)(42,46)(43,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(19,20)(24,28)(29,31)(30,32)(33,34)(35,40)(36,42)(37,43)(38,39)(41,44)(45,46)(47,48);(17,19,23,20)(18,24,27,28)(21,29,26,31)(22,30,25,32)(33,35)(34,40)(36,43)(37,42)(38,44)(39,41)(45,48)(46,47);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20,23,19)(18,24,27,28)(21,31,26,29)(22,30,25,32)(33,35)(34,41)(36,43)(37,46)(38,44)(39,40)(42,47)(45,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,26)(18,25,27,22)(19,29,20,31)(24,32,28,30)(33,36,38,45)(34,42,39,46)(35,43,44,48)(37,41,47,40);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,25)(18,26,27,21)(19,30,20,32)(24,31,28,29)(33,37,38,47)(34,43,39,48)(35,42,44,46)(36,41,45,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345012);