#242497

order	512
nilpotency class	3
isoclinism	isoclinic class of #233100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

772 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,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, 149693);