#239650

order	512
nilpotency class	3
isoclinism	isoclinic class of #233936
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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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,f^b=fi,g^b=gi,d^c=di,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, 146846);