#8750005

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	#52969	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=h,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, 8657201);