#118072

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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=hi,g^2=hi,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, 25268);