#348845

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×#1624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1027 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)(7,16)(8,12)(9,14)(10,15)(11,19)(13,18)(17,20)(21,22)(23,27)(24,32)(25,29)(26,30)(28,33)(31,36)(34,35);(1,2,3,4)(5,7,10,8)(6,12,15,16)(9,18,11,17)(13,14,20,19)(21,23,26,25)(22,27,30,29)(24,31,34,33)(28,32,36,35);(5,8,10,7)(6,13,15,20)(9,18,11,17)(12,14,16,19)(21,24,26,34)(22,28,30,36)(23,31,25,33)(27,32,29,35);(5,9,10,11)(6,14,15,19)(7,17,8,18)(12,13,16,20)(22,29)(23,25)(27,30)(28,35)(31,33)(32,36)

Polycyclic presentation:

2^9|b^2=ei,c^2=h,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=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, 256041);