#9685572

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,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=dh,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, 9592768);