#8687340

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	#54780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(33,34,39,44)(35,40,38,43)(36,42,37,41)(45,48,46,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(33,35,39,38)(34,40,44,43)(36,46,37,45)(41,48,42,47);(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,36,39,37)(34,41,44,42)(35,45,38,46)(40,47,43,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32)(33,34)(35,43)(36,41)(37,42)(38,40)(39,44)(45,48)(46,47);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,37,39,36)(34,42,44,41)(35,45,38,46)(40,47,43,48);(1,2)(3,6)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,38,39,35)(34,43,44,40)(36,46,37,45)(41,48,42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8594536);