#8096607

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(4,8,12,15)(11,13,16,14)(17,18)(19,22)(20,29)(21,31)(23,26)(24,27)(25,32)(28,30)(33,34,39,44)(35,38,43,40)(36,41,46,48)(37,47,45,42);(4,12)(8,15)(11,16)(13,14)(17,19)(18,20)(21,23)(22,27)(24,29)(25,26)(28,32)(30,31)(33,35)(34,40)(36,45)(37,46)(38,44)(39,43)(41,47)(42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,24,22)(18,19,27,29)(21,30,32,26)(23,25,28,31)(33,36,39,46)(34,41,44,48)(35,45,43,37)(38,42,40,47);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,21)(18,25)(19,28)(20,30)(22,26)(23,29)(24,32)(27,31)(33,37,39,45)(34,42,44,47)(35,46,43,36)(38,41,40,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,22,24,20)(18,19,27,29)(21,26,32,30)(23,25,28,31)(33,35)(34,38)(36,45)(37,46)(39,43)(40,44)(41,42)(47,48);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,15)(12,14)(17,23,24,28)(18,26,27,30)(19,21,29,32)(20,25,22,31)(33,38)(34,43)(35,44)(36,42)(37,41)(39,40)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gh,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, 8003803);