#116208

order	512
nilpotency class	3
isoclinism	isoclinic class of #10557
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

595 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)(3,9,12,6)(4,13)(5,7)(8,15)(10,14,16,11)(17,18,21,25,22,24,20,30)(19,26,29,32,28,23,27,31);(1,2)(3,10)(4,13)(5,7)(6,11)(8,15)(9,14)(12,16)(19,27,28,29)(23,31,26,32);(1,3)(2,6)(4,14)(5,12)(7,9)(8,16)(10,13)(11,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,26)(25,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fhi,d^c=dh,f^c=fi,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, 23404);