#9847622

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

4092 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(33,34,40,39)(35,43,44,38)(36,41,47,46)(37,48,45,42);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32)(33,35,40,44)(34,38,39,43)(36,42,47,48)(37,41,45,46);(2,9)(3,12)(4,14)(10,16)(18,23)(22,25)(24,31)(30,32)(33,36,40,47)(34,41,39,46)(35,42,44,48)(37,38,45,43);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,37,40,45)(34,42,39,48)(35,41,44,46)(36,38,47,43);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)(33,38)(34,35)(36,45)(37,47)(39,44)(40,43)(41,42)(46,48);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,39,40,34)(35,43,44,38)(36,46,47,41)(37,48,45,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=hi,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, 9754818);