#8750643

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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)(19,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(17,19)(18,23)(20,25)(21,28)(22,27)(24,32)(26,30)(29,31)(34,38)(39,41)(40,45)(46,47);(2,9)(4,14)(8,12)(11,13)(19,27)(20,29)(23,30)(24,28)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,20,22,29)(18,24,26,28)(19,25,27,31)(21,23,32,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,30)(28,29)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,21,22,32)(18,25,26,31)(19,28,27,24)(20,30,29,23)

Polycyclic presentation:

2^9|b^2=h,d^2=ghi,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, 8657839);