#239841

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1014 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)(7,15)(8,11)(9,13)(10,19)(12,17)(14,18)(16,20)(21,22,25,28)(23,26,30,35)(24,32,33,27)(29,36,31,34);(1,2)(3,4)(5,7,9,8)(6,11,13,15)(10,16,18,17)(12,14,20,19)(21,23)(22,26)(24,29)(25,30)(27,34)(28,35)(31,33)(32,36);(1,3,4,2)(5,8,9,7)(6,12,13,20)(10,16,18,17)(11,14,15,19)(23,29)(24,33)(26,34)(27,32)(30,31)(35,36);(7,8)(11,15)(12,20)(16,17)(23,30)(26,35)(29,31)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=dh,f^c=fh,g^c=gi

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, 147037);