#8821934

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	#50598	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729130);