#116823

order	512
nilpotency class	3
isoclinism	isoclinic class of #116822
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24019);