#118069

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,f^b=fh,d^c=di,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25265);