#420390

order	512
nilpotency class	3
isoclinism	isoclinic class of #420389
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15277
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327586);