#242708

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:

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

Polycyclic presentation:

2^9|a^2=efi,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, 149904);