#239828

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

967 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)(5,6,8,12)(7,14,15,10)(9,11,17,13)(16,19,20,18)(21,22)(23,27)(24,28)(25,34)(26,29)(30,33)(31,35)(32,36);(1,2)(3,4)(5,7)(6,10)(8,15)(9,16)(11,18)(12,14)(13,19)(17,20)(21,23)(22,27)(24,33)(25,31)(26,32)(28,30)(29,36)(34,35);(1,3,4,2)(6,11)(9,17)(10,18)(12,13)(14,19)(16,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,31)(32,35)(33,36);(7,15)(10,14)(16,20)(18,19)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,35,36,30)

Polycyclic presentation:

2^9|a^2=h,c^2=eh,d^2=i,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,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, 147024);