#118735

order	512
nilpotency class	3
isoclinism	isoclinic class of #118730
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

574 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,g^b=ghi,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25931);