#244109

order	512
nilpotency class	3
isoclinism	isoclinic class of #238019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=di,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, 151305);