#412557

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^2=h,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,h^a=hi,c^b=cg,e^c=ei,f^c=fi,g^c=gi

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, 319753);