#9863407

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9770603);