#117708

order	512
nilpotency class	3
isoclinism	isoclinic class of #117707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

658 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:

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

Polycyclic presentation:

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

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, 24904);