#413211

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

2414 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)(9,10,15,18)(11,20,23,32)(12,17,25,36)(13,27,14,19)(16,28,31,22)(21,33,37,40)(24,39,38,30)(26,35,29,34);(9,11,13,22,15,23,14,28)(10,16,19,20,18,31,27,32)(12,24,26,21,25,38,29,37)(17,33,34,39,36,40,35,30);(1,2,3,4)(9,12)(10,17)(11,21)(13,26)(14,29)(15,25)(16,30)(18,36)(19,34)(20,33)(22,38)(23,37)(24,28)(27,35)(31,39)(32,40);(5,6,7,8)(10,18)(12,25)(16,31)(19,27)(20,32)(21,37)(24,38)(26,29)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=di,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, 320407);