#390997

order	512
nilpotency class	3
isoclinism	isoclinic class of #32700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2760
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14596
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 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:

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

Polycyclic presentation:

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

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, 298193);