#374945

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

854 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)(3,8)(4,6)(5,7)(9,10,15,20,16,22,14,21)(11,23,26,18,12,27,25,17)(13,29,31,36,32,35,30,19)(24,34,39,40,28,33,38,37);(1,3,5,4)(2,6,7,8)(9,11,14,25,16,12,15,26)(10,17,20,23,22,18,21,27)(13,24,30,38,32,28,31,39)(19,33,29,37,36,34,35,40);(1,4,5,3)(2,6,7,8)(9,12)(10,18,22,17)(11,16)(13,28)(14,26)(15,25)(19,34,36,33)(20,27,21,23)(24,32)(29,40,35,37)(30,39)(31,38);(2,7)(6,8)(9,13,14,30,16,32,15,31)(10,19,20,29,22,36,21,35)(11,24,25,38,12,28,26,39)(17,33,23,37,18,34,27,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=eh,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282141);