#366404

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273600);