#262423

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

991 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,f^c=fh,g^c=gi

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, 169619);