#242701

order	512
nilpotency class	3
isoclinism	isoclinic class of #242697
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=di,g^b=gh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149897);