#397239

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304435);