#8195122

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=h,f^2=gi,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, 8102318);