#401466

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308662);