#8751150

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

3875 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)(3,11)(4,5)(6,8)(7,9)(10,15)(12,13)(14,16)(20,28)(24,31)(27,29)(30,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3,7,6)(2,8,9,11)(4,14,10,12)(5,13,15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(34,39)(35,41)(40,46)(42,47);(2,9)(4,10)(8,11)(12,14)(17,19,22,21)(18,23,25,26)(20,29,28,27)(24,32,31,30)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,4,7,10)(2,5,9,15)(3,12,6,14)(8,16,11,13)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5)(2,10)(3,13)(4,9)(6,16)(7,15)(8,12)(11,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48);(1,6,7,3)(2,8,9,11)(4,12,10,14)(5,13,15,16)(17,21,22,19)(18,23,25,26)(20,27,28,29)(24,32,31,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=gi,e^2=gh,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, 8658346);