#8112564

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

3994 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019760);