#512211

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,20,26)(15,22,28,40)(16,21,31,37)(17,24,34,36)(18,23,27,42)(19,35,33,25)(29,41,43,39)(30,44,32,38);(6,9)(10,12)(13,15,27,43,19,30,17,16,20,28,18,29,33,32,34,31)(14,21,36,44,25,39,23,22,26,37,24,38,35,41,42,40);(5,7,8,11)(6,10,9,12)(13,16,17,32,33,43,27,28,20,31,34,30,19,29,18,15)(14,22,23,41,35,44,36,37,26,40,42,39,25,38,24,21);(2,4)(15,28)(16,31)(21,37)(22,40)(29,43)(30,32)(38,44)(39,41)

Polycyclic presentation:

2^9|a^2=i,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, 419407);