#382869

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14345
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1127 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,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, 290065);