#349029

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
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×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256225);