#113140

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20336);