#116686

order	512
nilpotency class	3
isoclinism	isoclinic class of #116678
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

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

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, 23882);