#8821933

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	#51630	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|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, 8729129);