#430923

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 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:

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

Polycyclic presentation:

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

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, 338119);