#9062169

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	#46458	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969365);