#8096690

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

3973 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gh,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003886);