#411522

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=ef,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,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, 318718);