#382357

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=fgh,e^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289553);