#239488

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146684);