#9923918

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
prime radical	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3872 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)(4,6)(5,11)(7,9)(10,12)(13,16)(14,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,38,40)(35,41,43,36)(37,39,46,48)(42,45,47,44);(1,3)(2,8)(4,14)(5,12)(6,15)(7,13)(9,16)(10,11)(19,25)(22,29)(26,31)(30,32)(33,35,38,43)(34,36,40,41)(37,42,46,47)(39,44,48,45);(2,9)(3,13)(4,12)(11,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32);(1,4,7,12)(2,10,9,6)(3,14,13,5)(8,11,16,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,5,7,14)(2,11,9,15)(3,12,13,4)(6,8,10,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36)(34,35)(37,45)(38,41)(39,47)(40,43)(42,48)(44,46);(1,6,7,10)(2,12,9,4)(3,15,13,11)(5,16,14,8)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37)(34,39)(35,42)(36,44)(38,46)(40,48)(41,45)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,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, 9831114);