#120629

order	512
nilpotency class	3
isoclinism	isoclinic class of #10518
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	#1442
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

474 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27825);