#115669

order	512
nilpotency class	3
isoclinism	isoclinic class of #115666
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

892 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:

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

Polycyclic presentation:

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

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, 22865);