#265125

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3399	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172321);