#9236428

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3797 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(19,25)(20,27)(22,28)(23,30)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,48,47,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,4,6,7)(2,3,9,11)(5,13,15,12)(8,14,16,10)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,46,41,48);(1,5,6,15)(2,8,9,16)(3,10,11,14)(4,12,7,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,3)(2,4)(5,14)(6,11)(7,9)(8,13)(10,15)(12,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143624);