#8994522

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	#41257	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(34,38)(35,41)(36,44)(45,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,11)(12,14)(15,16)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(2,9)(3,5)(4,14)(6,16)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901718);