#119858

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

553 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27054);