#231476

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138672);