#370221

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
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×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

831 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,23,26,16)(13,18)(14,27)(17,19)(20,29,36,40)(21,34)(22,30)(24,37,39,32)(25,33,35,38)(28,31);(2,5)(6,7)(9,11,19,21,13,22,14,28)(10,15,17,34,18,30,27,31)(12,24,35,36,26,39,25,20)(16,32,33,29,23,37,38,40);(9,12,14,25,13,26,19,35)(10,16,17,33,18,23,27,38)(11,20,21,24,22,36,28,39)(15,29,31,32,30,40,34,37);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277417);