#512210

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33171	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,20)(15,26)(16,21)(17,28)(18,22)(23,27)(24,25);(6,9)(10,12)(13,14,21,15,18,25,28,27,19,23,17,24,22,26,16,20);(5,7,8,11)(6,10,9,12)(13,15,16,14,22,20,17,26,19,24,28,23,18,27,21,25);(2,4)(14,23)(15,24)(20,27)(25,26)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,h^a=hi,c^b=ci,d^b=di,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, 419406);