#117977

order	512
nilpotency class	3
isoclinism	isoclinic class of #10496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
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

561 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:

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

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,e^a=ehi,f^a=fhi,c^b=cf,e^b=eh,f^b=fh,d^c=di,f^c=fhi,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, 25173);