#318500

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

616 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=i,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,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, 225696);