#321890

order	512
nilpotency class	3
isoclinism	isoclinic class of #279051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

716 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229086);