#9912924

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

4073 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)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,37,40)(35,38,41,45)(36,39,44,48)(42,46,43,47);(2,6)(7,13)(8,11)(14,15)(19,25)(22,29)(26,31)(30,32)(35,41)(38,45)(42,43)(46,47);(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,44)(47,48);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,35)(34,38)(36,43)(37,41)(39,47)(40,45)(42,44)(46,48);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,48)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820120);