#10040727

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3801 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)(3,5)(4,10)(6,8)(7,9)(11,14)(12,15)(13,16)(17,18)(19,24)(20,28)(21,26)(22,30)(23,27)(25,31)(29,32)(33,34,39,40)(35,41,45,48)(36,42,46,37)(38,43,47,44);(1,3,7,6)(2,8,9,5)(4,12,13,11)(10,14,16,15)(17,19,23,22)(18,24,27,30)(20,29,31,26)(21,28,32,25)(34,40)(36,46)(41,48)(43,44);(2,9)(4,13)(5,8)(11,12)(17,20,23,31)(18,25,27,28)(19,26,22,29)(21,24,32,30)(33,35,39,45)(34,41,40,48)(36,44,46,43)(37,47,42,38);(1,4,7,13)(2,10,9,16)(3,11,6,12)(5,14,8,15)(17,20)(18,25)(19,26)(21,30)(22,29)(23,31)(24,32)(27,28)(33,36)(34,42)(35,44)(37,40)(38,41)(39,46)(43,45)(47,48);(1,5,7,8)(2,3,9,6)(4,14,13,15)(10,11,16,12)(17,21)(18,26)(19,25)(20,30)(22,28)(23,32)(24,31)(27,29)(33,37,39,42)(34,36,40,46)(35,38,45,47)(41,43,48,44);(1,6,7,3)(2,5,9,8)(4,12,13,11)(10,14,16,15)(17,22,23,19)(18,24,27,30)(20,29,31,26)(21,25,32,28)(33,38)(34,43)(35,37)(36,41)(39,47)(40,44)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9947923);