#8463762

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=hi,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, 8370958);