#345842

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253038);