#118144

order	512
nilpotency class	3
isoclinism	isoclinic class of #113356
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,e^b=ehi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25340);