#345897

order	512
nilpotency class	3
isoclinism	isoclinic class of #345888
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

548 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:

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

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=defi,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253093);