#8496897

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8404093);