#212767

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 119963);