#8650334

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

3958 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)(4,5)(6,11)(7,9)(10,16)(12,15)(13,14)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,30,32,31)(33,34,40,44)(35,38,39,41)(36,43,37,42)(45,46,47,48);(1,3)(2,8)(4,11)(5,6)(7,12)(9,15)(10,13)(14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,40,39)(34,41,44,38)(36,45,37,47)(42,46,43,48);(2,9)(3,12)(5,16)(11,13)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,36,40,37)(34,42,44,43)(35,45,39,47)(38,48,41,46);(1,4,7,10)(2,5,9,16)(3,13,12,11)(6,8,14,15)(17,20,22,18)(19,25,27,23)(21,24,29,28)(26,30,32,31)(33,37,40,36)(34,43,44,42)(35,45,39,47)(38,48,41,46);(1,5)(2,10)(3,14)(4,9)(6,12)(7,16)(8,11)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38,40,41)(34,35,44,39)(36,46,37,48)(42,47,43,45);(1,6,7,14)(2,11,9,13)(3,5,12,16)(4,15,10,8)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,39,40,35)(34,41,44,38)(36,47,37,45)(42,46,43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8557530);