#242702

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

607 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=h,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, 149898);