#8691822

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	#36901	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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)(3,8)(4,10)(5,14)(6,12)(7,13)(9,15)(11,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(1,3,7,6)(2,8,13,12)(4,11,15,14)(5,10,16,9)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32)(34,40)(35,44)(36,48)(42,46);(1,4)(2,9)(3,14)(5,12)(6,11)(7,15)(8,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,4)(2,10)(3,11)(5,12)(6,14)(7,15)(8,16)(9,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,5)(2,11)(3,10)(4,12)(6,9)(7,16)(8,15)(13,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,6,7,3)(2,12,13,8)(4,11,15,14)(5,10,16,9)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8599018);