#8581503

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

3884 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=hi,e^2=ghi,f^2=i,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, 8488699);