#450124

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15958
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 357320);