#238733

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1039 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)(5,6,9,13)(7,10,14,19)(8,16,18,12)(11,17,20,15)(21,22,26,30)(23,27,33,36)(24,28,25,29)(31,34,32,35);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,16)(12,20)(13,19)(17,18)(21,23)(22,27)(24,32)(25,31)(26,33)(28,35)(29,34)(30,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,14)(10,16)(12,19)(13,20)(15,18)(22,28)(23,31)(24,25)(29,30)(32,33)(34,35);(7,14)(10,19)(11,20)(15,17)(21,24,26,25)(22,28,30,29)(23,31,33,32)(27,34,36,35)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,g^b=gi,d^c=dh,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, 145929);