#112376

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

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,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, 19572);