#8534915

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

3939 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=hi,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, 8442111);