#412113

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1360 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)(11,23,27,32)(12,19,30,35)(13,18,29,36)(14,21,15,20)(17,26,33,25)(24,31,39,40)(28,37,38,34);(5,6)(7,8)(9,11)(10,17)(12,28)(13,24)(14,26)(15,25)(16,27)(18,34)(19,31)(20,32)(21,23)(22,33)(29,39)(30,38)(35,40)(36,37);(2,4)(6,8)(9,12,14,29,16,30,15,13)(10,18,20,35,22,36,21,19)(11,24,25,38,27,39,26,28)(17,31,23,37,33,40,32,34);(9,13,15,30,16,29,14,12)(10,19,21,36,22,35,20,18)(11,24,25,38,27,39,26,28)(17,31,23,37,33,40,32,34)

Polycyclic presentation:

2^9|a^2=i,c^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^b=ei,f^b=fi,g^b=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, 319309);