#109203

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16399);