#377988

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1185 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,h^c=hi

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, 285184);