#495366

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
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×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

848 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)(9,10,14,19,28,32,30,18,15,20,27,38,12,17,13,29)(11,21,24,16,22,34,23,33,26,39,40,36,25,37,31,35);(2,5)(6,7)(9,11,12,22,15,26,28,25)(10,16,32,35,20,36,17,33)(13,24,27,23,30,40,14,31)(18,37,38,39,29,34,19,21);(1,3,4,8)(2,6,5,7);(9,12,15,28)(10,17,20,32)(11,22,26,25)(13,27,30,14)(16,33,36,35)(18,19,29,38)(21,34,39,37)(23,40,31,24)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=f,d^2=i,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=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, 402562);