#8437802

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

3959 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)(19,23)(20,30)(21,25)(22,26)(24,27)(28,32)(29,31)(34,39)(35,42)(41,46)(43,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,24,22)(18,23,27,26)(20,29,21,28)(25,32,30,31)(33,34,38,39)(35,37,42,40)(36,41,45,46)(43,44,48,47);(17,20,24,21)(18,25,27,30)(19,28,22,29)(23,31,26,32)(33,35)(34,37)(36,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,21,24,20)(18,25,27,30)(19,29,22,28)(23,31,26,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,46)(45,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,22,24,19)(18,26,27,23)(20,29,21,28)(25,32,30,31)(33,36,38,45)(34,41,39,46)(35,43,42,48)(37,44,40,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,23,24,26)(18,19,27,22)(20,31,21,32)(25,28,30,29)(33,37,38,40)(34,42,39,35)(36,44,45,47)(41,48,46,43)

Polycyclic presentation:

2^9|b^2=gi,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, 8344998);