#239760

order	512
nilpotency class	3
isoclinism	isoclinic class of #233984
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 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)(5,6,10,13)(7,14,16,19)(8,12,9,17)(11,18,20,15)(21,22,25,29)(23,26,31,34)(24,27,33,28)(30,35,32,36);(1,2)(3,4)(21,23)(22,26)(24,30)(25,31)(27,35)(28,36)(29,34)(32,33);(1,3,4,2)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(22,27)(23,30)(24,33)(28,29)(31,32)(35,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(23,31)(26,34)(30,32)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146956);