#8661377

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,12)(4,10)(5,7)(6,9)(8,11)(13,15)(14,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,8,6)(2,9,11,12)(4,13,5,14)(7,15,10,16)(18,23)(19,26)(25,28)(27,32)(35,42)(39,46)(43,47)(44,48);(1,4,8,5)(2,10,11,7)(3,13,6,14)(9,16,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,5,8,4)(2,10,11,7)(3,14,6,13)(9,16,12,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6,8,3)(2,9,11,12)(4,14,5,13)(7,15,10,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,7,8,10)(2,5,11,4)(3,15,6,16)(9,14,12,13)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8568573);