#117770

order	512
nilpotency class	3
isoclinism	isoclinic class of #10018
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	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

599 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:

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

Polycyclic presentation:

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

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, 24966);