#8701653

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

4030 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608849);