#378761

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	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 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)(3,7)(4,6)(5,8)(9,10,16,21,14,20,15,22)(11,23,13,31,25,17,26,19)(12,27,30,18,29,36,32,35)(24,34,28,37,38,40,39,33);(1,3,4,8)(2,5,6,7)(9,11,16,13,14,25,15,26)(10,17,21,19,20,23,22,31)(12,28,32,24,29,39,30,38)(18,34,27,33,35,40,36,37);(9,12)(10,18)(11,24)(13,28)(14,29)(15,32)(16,30)(17,33)(19,34)(20,35)(21,36)(22,27)(23,37)(25,38)(26,39)(31,40);(2,6)(5,7)(9,13,16,25,14,26,15,11)(10,19,21,23,20,31,22,17)(12,28,30,38,29,39,32,24)(18,34,36,37,35,40,27,33)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=h,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, 285957);