#321605

order	512
nilpotency class	3
isoclinism	isoclinic class of #278091
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×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

709 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228801);