#506534

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#25777	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

827 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)(7,11)(8,10)(9,12)(13,14,18,24,17,23,31,41,20,26,27,36,19,25,35,42)(15,21,28,37,33,40,44,39,29,38,34,22,16,30,32,43);(5,7,8,12)(6,9,10,11)(13,15,27,34,17,33,35,32,20,29,18,28,19,16,31,44)(14,21,36,22,23,40,42,43,26,38,24,37,25,30,41,39);(2,4)(13,16,20,33)(14,22,26,37)(15,17,29,19)(18,34,27,28)(21,24,38,36)(23,39,25,43)(30,41,40,42)(31,44,35,32);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,16,29,33)(18,31,27,35)(21,30,38,40)(22,39,37,43)(24,41,36,42)(28,32,34,44)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413730);