#364389

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2468 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)(9,10,14,19,15,20,13,18)(11,16,22,30,23,31,27,35)(12,17,26,34,25,33,28,36)(21,29,38,40,37,39,24,32);(2,4)(9,11,14,22,15,23,13,27)(10,16,19,30,20,31,18,35)(12,24,26,21,25,38,28,37)(17,32,34,29,33,40,36,39);(6,8)(9,12)(10,17)(11,21)(13,28)(14,26)(15,25)(16,29)(18,36)(19,34)(20,33)(22,38)(23,37)(24,27)(30,40)(31,39)(32,35);(2,4)(12,25)(17,33)(21,37)(24,38)(26,28)(29,39)(32,40)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271585);