#239839

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1084 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)(7,10)(8,16)(9,13)(11,15)(12,18)(14,19)(17,20)(21,22,25,28)(23,26,30,35)(24,32,33,27)(29,36,31,34);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,16)(12,20)(13,19)(17,18)(21,23)(22,26)(24,29)(25,30)(27,34)(28,35)(31,33)(32,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,14)(10,16)(12,19)(13,20)(15,18)(23,29)(24,33)(26,34)(27,32)(30,31)(35,36);(7,14)(10,19)(11,20)(15,17)(23,30)(26,35)(29,31)(34,36)

Polycyclic presentation:

2^9|a^2=i,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, 147035);