#9936329

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

4272 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,e^2=g,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, 9843525);