#364816

order	512
nilpotency class	3
isoclinism	isoclinic class of #196161
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1229 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)(9,10,14,20,15,19,13,18)(11,16,22,31,23,32,29,35)(12,24,27,17,28,34,26,33)(21,36,37,40,25,39,38,30);(2,5)(6,7)(9,11,14,22,15,23,13,29)(10,16,20,31,19,32,18,35)(12,25,27,38,28,21,26,37)(17,30,34,36,33,40,24,39);(9,12,15,28)(10,17,19,33)(11,21,23,25)(13,27,14,26)(16,30,32,40)(18,34,20,24)(22,38,29,37)(31,39,35,36);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 272012);