#8709145

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

4147 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)(4,12,13,9)(11,15,14,16)(33,34,38,42)(35,37,41,39)(36,44,46,40)(43,45,48,47);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(4,13)(9,12)(11,14)(15,16)(18,23)(19,25)(24,31)(26,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,45,44,47);(1,4)(2,9)(3,11)(5,14)(6,12)(7,13)(8,15)(10,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,45)(44,47)(46,48);(1,5,7,3)(2,8,6,10)(4,14,13,11)(9,15,12,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,47,39,45);(1,6,7,2)(3,10,5,8)(4,9,13,12)(11,15,14,16)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8616341);