#8096521

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34,39,42)(35,36,37,40)(38,41,46,48)(43,44,45,47);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(20,28)(24,31)(27,32)(29,30)(33,35,39,37)(34,40,42,36)(38,43,46,45)(41,47,48,44);(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,39,35)(34,40,42,36)(38,43,46,45)(41,44,48,47);(1,5)(2,8)(3,9)(4,13)(6,10)(7,15)(11,12)(14,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,46)(34,41,42,48)(35,43,37,45)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^2=i,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, 8003717);