#363559

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×#380
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

978 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,h^c=hi

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, 270755);