#115279

order	512
nilpotency class	3
isoclinism	isoclinic class of #115274
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22475);