#8114390

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	#53695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,14)(15,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,2,5,8)(3,6,10,9)(4,7,13,12)(11,14,16,15)(18,23)(19,26)(25,28)(27,32)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(3,10)(6,9)(11,16)(14,15)(17,18,22,23)(19,24,26,21)(20,25,30,28)(27,31,32,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3,5,10)(2,6,8,9)(4,11,13,16)(7,14,12,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021586);