#413198

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2326 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:

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

Polycyclic presentation:

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

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, 320394);