#481544

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

928 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^e=fi

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, 388740);