#367799

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

848 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)(9,10,15,20,16,21,14,31)(11,22,24,33,25,19,13,17)(12,26,29,36,30,35,28,18)(23,38,39,34,40,32,27,37);(1,3,5,4)(2,6,7,8)(9,11,15,24,16,25,14,13)(10,17,31,19,21,33,20,22)(12,27,28,40,30,39,29,23)(18,34,26,32,36,37,35,38);(1,4,5,3)(2,6,7,8)(9,12,14,28,16,30,15,29)(10,18,20,26,21,36,31,35)(11,23,13,27,25,40,24,39)(17,32,22,37,33,38,19,34);(2,7)(6,8)(9,13,14,25,16,24,15,11)(10,19,20,17,21,22,31,33)(12,27,28,40,30,39,29,23)(18,34,26,32,36,37,35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=e,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^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, 274995);