#129548

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3480	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=ei,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36744);