#345922

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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 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)(17,18,21,23)(19,22,26,30)(20,24,27,31)(25,29,28,32);(1,2,4,6,5,9,14,8)(3,11,10,15,13,7,12,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(1,3,5,13)(2,7,9,11)(4,12,14,10)(6,15,8,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(2,8)(4,14)(6,9)(7,16)(10,12)(11,15)(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=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, 253118);