#425084

order	512
nilpotency class	3
isoclinism	isoclinic class of #425084
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15328
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 332280);