#110609

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

724 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

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, 17805);