#242490

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

718 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)(21,22,26,34,25,30,24,29)(23,27,31,28,33,36,32,35);(1,2)(3,4)(5,7,9,17,11,8,10,18)(6,12,14,13,15,20,16,19)(21,23)(22,27)(24,32)(25,33)(26,31)(28,34)(29,35)(30,36);(2,4)(5,8,11,7)(6,13,15,19)(9,17,10,18)(12,16,20,14)(21,23)(22,28)(24,31)(25,33)(26,32)(27,34)(29,36)(30,35);(21,24,25,26)(22,29,30,34)(23,31,33,32)(27,28,36,35)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=h,d^2=i,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,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, 149686);