#242704

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

626 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:

(1,2,5,10,6,11,4,9)(3,7,12,15,13,16,14,8)(17,18,22,25)(19,26,24,21)(20,27,31,28)(23,29,32,30);(1,3)(2,7)(4,14)(5,12)(6,13)(8,9)(10,15)(11,16)(17,19,21,18,22,24,26,25)(20,28,30,23,31,27,29,32);(1,3)(2,8)(4,12)(5,14)(6,13)(7,9)(10,16)(11,15)(17,20,22,31)(18,23,25,32)(19,27,24,28)(21,29,26,30);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,15,16,8)(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=h,d^2=i,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, 149900);