#242560

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

708 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=i,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149756);