#8096522

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	#46103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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)(3,10)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34,39,43)(35,37,38,40)(36,42,47,41)(44,48,45,46);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(20,28)(24,31)(27,32)(29,30)(33,35,39,38)(34,40,43,37)(36,45,47,44)(41,46,42,48);(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,48)(42,43);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,40)(34,35,43,38)(36,46,47,48)(41,44,42,45);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36)(34,42)(35,45)(37,46)(38,44)(39,47)(40,48)(41,43);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,35)(34,40,43,37)(36,45,47,44)(41,48,42,46)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,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, 8003718);