#117607

order	512
nilpotency class	3
isoclinism	isoclinic class of #117607
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24803);