#9945512

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

3952 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,32,28,31)(33,34,39,41)(35,44,38,40)(36,42,46,48)(37,43,47,45);(2,7)(3,10)(4,12)(13,14)(17,18,21,22)(19,23,26,30)(20,24,29,27)(25,31,28,32)(33,35,39,38)(34,40,41,44)(36,45,46,43)(37,42,47,48);(18,22)(23,30)(24,27)(31,32)(34,41)(36,46)(40,44)(43,45);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,13,15,14)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)(33,36)(34,42)(35,43)(37,40)(38,45)(39,46)(41,48)(44,47);(1,4,6,12)(2,9,7,15)(3,11,10,16)(5,14,8,13)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31)(33,37)(34,43)(35,42)(36,40)(38,48)(39,47)(41,45)(44,46);(1,5,6,8)(2,10,7,3)(4,13,12,14)(9,11,15,16)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32)(33,38,39,35)(34,44,41,40)(36,45,46,43)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852708);