#8750193

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	#42838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,9)(4,14)(8,12)(11,13)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=gh,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, 8657389);