#110104

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

660 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)(7,11,14,19)(8,15,16,18)(9,13)(10,12)(17,20)(21,22)(23,29,31,36)(24,28)(25,27)(26,30,34,32)(33,35);(1,2,3,4)(5,7)(6,11)(8,12)(9,14)(10,18)(13,19)(15,17)(16,20)(21,23)(22,26)(24,32)(25,31)(27,34)(28,36)(29,35)(30,33);(5,8,10,19,9,16,17,11)(6,7,12,18,13,14,20,15)(22,27)(24,33)(26,34)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17300);