#8535212

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
prime radical	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=ghi,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, 8442408);