#8662818

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	#42266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570014);