#8661362

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

4191 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^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, 8568558);