#8475078

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

4546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382274);