#8750130

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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)(3,8,6,10)(4,9,15,5)(12,14,13,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,6)(2,8,11,10)(4,13,15,12)(5,14,9,16)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4,7,15)(2,9,11,5)(3,12,6,13)(8,14,10,16)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,5)(3,13)(6,12)(7,15)(8,14)(9,11)(10,16)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5)(2,4)(3,14)(6,16)(7,9)(8,13)(10,12)(11,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47);(1,6,7,3)(2,10,11,8)(4,13,15,12)(5,14,9,16)(17,21,22,19)(18,23,25,26)(20,27,29,28)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8657326);