#8425647

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332843);