#8096840

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	#52486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=h,f^2=g,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, 8004036);