#117213

order	512
nilpotency class	3
isoclinism	isoclinic class of #117213
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

680 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:

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

Polycyclic presentation:

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

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, 24409);