#119111

order	512
nilpotency class	3
isoclinism	isoclinic class of #119110
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	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26307);