#8750250

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3893 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)(17,18,23,24)(19,28,22,25)(20,26,30,32)(21,29,31,27)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3,7,6)(2,8,11,10)(4,13,15,12)(5,14,9,16)(18,24)(20,30)(25,28)(27,29)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,15)(2,9,11,5)(3,12,6,13)(8,14,10,16)(17,19,23,22)(18,25,24,28)(20,27,30,29)(21,26,31,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4)(2,5)(3,13)(6,12)(7,15)(8,14)(9,11)(10,16)(17,20,23,30)(18,26,24,32)(19,29,22,27)(21,28,31,25)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,4)(3,14)(6,16)(7,9)(8,13)(10,12)(11,15)(17,21)(18,27)(19,26)(20,28)(22,32)(23,31)(24,29)(25,30)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48);(1,6,7,3)(2,10,11,8)(4,13,15,12)(5,14,9,16)(17,22,23,19)(18,25,24,28)(20,29,30,27)(21,26,31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,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, 8657446);