#380134

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1165 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:

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

Polycyclic presentation:

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

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, 287330);