#321413

order	512
nilpotency class	3
isoclinism	isoclinic class of #298661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228609);