#348812

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
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×D₄²	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1161 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256008);