#8095859

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	#46264	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,34,39,42)(35,37,44,40)(36,45,38,41)(43,48,46,47);(4,12)(7,14)(10,16)(13,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,40)(36,46)(37,42)(38,43)(39,44)(41,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,36,39,38)(34,41,42,45)(35,43,44,46)(37,47,40,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37)(34,35)(36,47)(38,48)(39,40)(41,43)(42,44)(45,46);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,35)(34,37)(36,43)(38,46)(39,44)(40,42)(41,47)(45,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,39,36)(34,41,42,45)(35,43,44,46)(37,48,40,47)

Polycyclic presentation:

2^9|a^2=g,c^2=g,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, 8003055);