#8395038

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	#50821	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(33,34,39,43)(35,42,37,40)(36,41,46,38)(44,48,47,45);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(35,37)(40,42)(44,47)(45,48);(4,12)(7,14)(10,16)(13,15)(18,23)(21,29)(24,25)(27,32)(33,35,39,37)(34,40,43,42)(36,44,46,47)(38,48,41,45);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36)(34,41)(35,44)(37,47)(38,43)(39,46)(40,45)(42,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,35)(34,42,43,40)(36,44,46,47)(38,48,41,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,41)(34,36,43,46)(35,45,37,48)(40,47,42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8302234);