#321429

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

756 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)(5,6,8,11,9,12,17,20)(7,13,15,18,14,19,16,10)(21,22)(23,29)(24,27)(25,28)(26,32)(30,34)(31,35)(33,36);(1,2,4,3)(7,14)(10,18)(13,19)(15,16)(21,23,30,31,24,32,25,33)(22,26,34,36,27,29,28,35);(5,7,9,14)(6,10,12,18)(8,16,17,15)(11,19,20,13)(22,27)(23,31)(25,30)(26,35)(29,36)(32,33);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=h,d^2=ei,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, 228625);