#407669

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

956 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)(3,5)(4,6)(7,8)(9,10,13,17,14,18,26,32)(11,19,21,27,22,31,25,15)(12,16,24,30,23,29,36,39)(20,33,35,38,34,40,37,28);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,31)(18,27)(19,32)(21,26)(23,34)(24,37)(29,38)(30,40)(33,39)(35,36);(12,23)(15,27)(16,29)(19,31)(20,34)(24,36)(30,39)(35,37);(2,6)(5,8)(9,12,13,24,14,23,26,36)(10,16,17,30,18,29,32,39)(11,20,25,37,22,34,21,35)(15,28,31,40,27,38,19,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314865);