#507161

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

620 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:

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

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=fi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414357);