#10040518

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9947714);