#8197192

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	#47589	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:

(1,2,7,10)(3,4,5,8)(6,9,14,16)(11,12,13,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,30)(31,32);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34)(35,37)(36,41)(38,39)(40,43)(42,44)(45,46)(47,48);(6,14)(9,16)(11,13)(12,15)(18,23)(22,25)(24,28)(30,31)(34,39)(37,40)(41,46)(44,47);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,20,21,29)(18,24,23,28)(19,26,27,32)(22,30,25,31)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,46)(45,48);(1,5,7,3)(2,8,10,4)(6,13,14,11)(9,15,16,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,36,38,45)(34,41,39,46)(35,42,43,48)(37,44,40,47);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,29)(31,32)(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=gh,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, 8104388);