#345962

order	512
nilpotency class	3
isoclinism	isoclinic class of #345928
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

534 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)(3,9)(4,13)(8,15)(11,12)(14,16)(17,18,21,23)(19,22,26,30)(20,24,27,31)(25,29,28,32);(1,2,4,6,5,8,13,15)(3,10,9,14,12,7,11,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,12)(2,7,8,10)(4,11,13,9)(6,14,15,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(2,6)(4,13)(7,14)(8,15)(9,11)(10,16)(18,23)(22,30)(24,31)(29,32)

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=defi,e^a=eh,f^a=fh,c^b=cg,d^b=deh,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, 253158);