#110709

order	512
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

684 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)(5,6,10,14,11,16,9,15)(7,17,18,8,19,20,13,12)(21,22)(23,29,31,36)(24,28)(25,27)(26,30,34,32)(33,35);(5,7,9,13,11,19,10,18)(6,12,14,17,16,8,15,20)(21,23)(22,26)(24,32)(25,31)(27,34)(28,36)(29,35)(30,33);(1,2,3,4)(5,8)(6,13)(7,15)(9,20)(10,17)(11,12)(14,19)(16,18)(22,27)(24,33)(26,34)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di,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, 17905);