#8750407

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32)(33,34,39,42)(35,37,44,41)(36,45,38,40)(43,47,46,48);(2,7)(3,11)(9,12)(10,16)(17,19)(18,23)(20,25)(21,28)(22,27)(24,32)(26,30)(29,31)(33,35)(34,37)(36,46)(38,43)(39,44)(40,47)(41,42)(45,48);(2,7)(3,11)(9,12)(10,16)(19,27)(20,29)(23,30)(24,28)(33,36,39,38)(34,40,42,45)(35,43,44,46)(37,48,41,47);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,20,22,29)(18,24,26,28)(19,25,27,31)(21,23,32,30)(33,37,39,41)(34,35,42,44)(36,47,38,48)(40,46,45,43);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,30)(28,29)(33,35)(34,41)(36,43)(37,42)(38,46)(39,44)(40,47)(45,48);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,21,22,32)(18,25,26,31)(19,28,27,24)(20,30,29,23)(33,38,39,36)(34,40,42,45)(35,43,44,46)(37,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8657603);