#113579

order	512
nilpotency class	3
isoclinism	isoclinic class of #113576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

814 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20775);