#253330

order	512
nilpotency class	3
isoclinism	isoclinic class of #253315
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 160526);