#117720

order	512
nilpotency class	3
isoclinism	isoclinic class of #10500
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

628 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)(9,10,14,18,15,19,13,28)(11,20,24,31,23,36,22,16)(12,17,26,34,27,33,29,32)(21,35,37,40,25,39,38,30);(9,11,13,22,15,23,14,24)(10,16,18,20,19,31,28,36)(12,25,26,38,27,21,29,37)(17,30,32,39,33,40,34,35);(1,2)(3,7)(4,6)(5,8)(9,12,15,27)(10,17,19,33)(11,21,23,25)(13,29,14,26)(16,30,31,40)(18,34,28,32)(20,35,36,39)(22,38,24,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24916);