#8097837

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8005033);