#116101

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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

645 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:

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

Polycyclic presentation:

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

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, 23297);