#116665

order	512
nilpotency class	3
isoclinism	isoclinic class of #116660
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(2,7)(3,8)(4,5)(9,10,14,18,15,20,13,19)(11,21,24,31,25,35,23,16)(12,26,27,17,29,34,28,33)(22,30,37,40,38,32,39,36);(1,2,4,7,6,3,5,8)(9,11,13,23,15,25,14,24)(10,16,18,21,20,31,19,35)(12,22,27,37,29,38,28,39)(17,32,26,40,33,30,34,36);(1,3,6,2)(4,8,5,7)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,26)(19,34)(20,33)(21,36)(23,37)(24,39)(25,38)(31,32)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=eh,f^c=fi,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, 23861);