#115524

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

901 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,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, 22720);