#415518

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=fi,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322714);