#8197190

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	#48304	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

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

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, 8104386);