#411198

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

670 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)(3,5)(4,6)(7,8)(9,10,15,27,14,20,13,19)(11,21,23,31,24,17,28,16)(12,18,29,38,26,34,25,33)(22,35,36,40,37,32,39,30);(1,3,4,7)(2,5,6,8)(9,11)(10,16,20,31)(12,22)(13,23)(14,24)(15,28)(17,19,21,27)(18,30,34,40)(25,36)(26,37)(29,39)(32,33,35,38);(9,11)(10,17,20,21)(12,22)(13,23)(14,24)(15,28)(16,27,31,19)(18,32,34,35)(25,36)(26,37)(29,39)(30,38,40,33);(2,6)(5,8)(9,12,13,25,14,26,15,29)(10,18,19,33,20,34,27,38)(11,22,23,36,24,37,28,39)(16,30,21,35,31,40,17,32)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=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, 318394);