#8718247

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8625443);