#112375

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

949 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6,9,11,8,7,12,10)(13,14)(15,24)(16,29,32,38)(17,33)(18,22)(19,23)(20,26)(21,31,36,30)(25,34,42,44)(27,35)(28,37)(39,41,43,40);(1,2)(3,4)(9,12)(10,11)(13,15,19,28)(14,20,23,35)(16,25,32,42)(17,27,18,26)(21,34,36,44)(22,37,33,24)(29,39,38,43)(30,41,31,40);(2,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,31)(18,30)(19,32)(20,34)(22,38)(23,36)(24,39)(26,41)(27,40)(28,42)(29,33)(35,44)(37,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=eh,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, 19571);