#115596

order	512
nilpotency class	3
isoclinism	isoclinic class of #10192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1051 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)(9,10,14,18,15,20,13,19)(11,21,23,16,24,31,29,32)(12,25,26,17,28,35,27,34)(22,33,38,40,39,30,37,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,33)(18,31)(19,21)(20,32)(25,36)(26,38)(27,37)(28,39)(30,34)(35,40);(2,6)(3,4)(7,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,30)(18,25)(19,35)(20,34)(21,36)(23,37)(24,39)(29,38)(31,40)(32,33)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,e^b=ehi,f^b=fi,d^c=di,e^c=ehi,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, 22792);