#239229

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146425);