#242622

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
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	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,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,d^b=di,d^c=dh,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, 149818);