#242654

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
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

716 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=i,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,d^b=dhi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

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, 149850);