#323015

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
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	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

683 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:

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

Polycyclic presentation:

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

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, 230211);