#345817

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	#24176	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

555 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)(19,24)(20,28)(21,23)(22,25)(26,32)(27,30)(29,31);(1,2,5,9,6,10,4,7)(3,12,11,16,13,8,14,15)(18,23)(19,26)(20,22)(24,31)(27,29)(30,32);(1,3)(2,8)(4,11)(5,14)(6,13)(7,15)(9,16)(10,12)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(2,9)(4,5)(7,10)(8,16)(11,14)(12,15)(17,20,21,22)(18,25,23,28)(19,26,27,29)(24,31,30,32)

Polycyclic presentation:

2^9|b^2=f,d^2=hi,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, 253013);