#412558

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=hi,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, 319754);