#369786

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(3,7)(4,5)(6,8)(9,10,14,20,16,13,15,19)(11,21,25,23,26,34,24,17)(12,27,31,18,32,36,30,29)(22,37,28,40,39,33,38,35);(2,5)(6,7)(9,11,14,25,16,26,15,24)(10,17,20,21,13,23,19,34)(12,28,31,39,32,38,30,22)(18,35,36,37,29,40,27,33);(9,12,15,30,16,32,14,31)(10,18,19,27,13,29,20,36)(11,22,25,28,26,39,24,38)(17,33,21,35,23,37,34,40);(1,3,4,8)(2,6,5,7)(9,13)(10,16)(11,23)(12,29)(14,19)(15,20)(17,26)(18,32)(21,24)(22,37)(25,34)(27,31)(28,40)(30,36)(33,39)(35,38)

Polycyclic presentation:

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

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, 276982);