#9026922

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

(1,2,7,4)(3,8,11,12)(5,9,15,13)(6,14,16,10)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,48,47,46);(1,3)(2,8)(4,12)(5,14)(6,13)(7,11)(9,16)(10,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(2,4)(3,11)(5,15)(6,16)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,4,7,2)(3,12,11,8)(5,9,15,13)(6,14,16,10)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,46,41,48);(1,5)(2,9)(3,10)(4,13)(6,8)(7,15)(11,14)(12,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,6,7,16)(2,10,4,14)(3,13,11,9)(5,8,15,12)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(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=g,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934118);