#505196

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	C₂³×D₈	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

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=f,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, 412392);