#10247180

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	#49972	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154376);