#473890

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381086);