#242367

order	512
nilpotency class	3
isoclinism	isoclinic class of #242365
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

670 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149563);