#8095124

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	#53128	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4342 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002320);