#9935860

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

4268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 9843056);