#278748

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 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)(5,6,9,13,8,12,17,11)(7,14,16,19,15,10,18,20)(21,22)(23,26)(24,27)(25,28)(29,36)(30,34)(31,33)(32,35);(1,2,4,3)(5,7,8,15)(6,10,12,14)(9,16,17,18)(11,19,13,20)(21,23,24,30)(22,26,27,34)(25,29,32,31)(28,36,35,33);(6,11)(9,17)(10,19)(12,13)(14,20)(16,18)(22,27)(23,29)(25,32)(26,33)(30,31)(34,36);(1,3,4,2)(6,12)(7,15)(11,13)(16,18)

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=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, 185944);