#348831

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 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)(7,13)(8,15)(10,17)(12,19)(14,16)(18,20)(21,22,26,30)(23,27,25,29)(24,32,34,35)(28,31,36,33);(1,2,3,4)(5,6,9,11)(7,10,14,18)(8,12,15,19)(13,17,16,20)(21,23,26,25)(22,27,30,29)(24,31,34,33)(28,32,36,35);(5,7,9,14)(6,10,11,18)(8,16,15,13)(12,20,19,17)(21,24,26,34)(22,28,30,36)(23,31,25,33)(27,32,29,35);(6,11)(10,18)(12,19)(17,20)(22,29)(23,25)(27,30)(28,35)(31,33)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=h,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, 256027);