#8436670

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

3974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343866);