#8455722

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,11,14,8)(4,15,13,6)(5,16,12,9)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,3)(2,8)(4,12)(5,13)(6,9)(7,14)(10,11)(15,16)(18,23)(19,26)(25,28)(27,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,4,7,13)(2,6,10,15)(3,12,14,5)(8,9,11,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5,7,12)(2,9,10,16)(3,13,14,4)(6,11,15,8)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,6)(2,4)(3,9)(5,11)(7,15)(8,12)(10,13)(14,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362918);