#392917

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,d^b=di

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, 300113);