#8534805

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

4116 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)(3,13)(4,10)(5,7)(6,11)(8,12)(9,16)(14,15)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,32,30,28);(1,3)(2,9)(4,14)(5,11)(6,10)(7,15)(8,16)(12,13)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,27)(31,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(1,4,8,5)(2,10,12,7)(3,14,16,11)(6,13,15,9)(17,19,22,29)(18,24,26,23)(20,27,25,30)(21,28,31,32)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,5,8,4)(2,10,12,7)(3,11,16,14)(6,13,15,9)(17,20)(18,21)(19,27)(22,25)(23,32)(24,28)(26,31)(29,30)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,6,8,15)(2,11,12,14)(3,10,16,7)(4,9,5,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,30)(31,32)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,7,8,10)(2,5,12,4)(3,15,16,6)(9,11,13,14)(17,21)(18,25)(19,28)(20,26)(22,31)(23,27)(24,30)(29,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442001);