#505216

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

749 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)(7,10)(8,9)(11,12)(13,14,20,27)(15,29,33,44)(16,34,35,24)(17,23,36,43)(18,37,31,26)(19,39,38,25)(21,40,30,28)(22,41,42,32);(1,2)(3,4)(6,9)(10,12)(13,15,19,16,21,17,18,32,20,33,38,35,30,36,31,41)(14,22,26,23,28,34,25,29,27,42,37,43,40,24,39,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,31,15,30,41,38,36,20,35,18,33,21,32,19,17)(14,23,39,22,40,44,37,24,27,43,25,42,28,29,26,34);(6,9)(10,12)(13,17,19,32,21,33,18,35,20,36,38,41,30,15,31,16)(14,24,26,44,28,22,25,23,27,34,37,29,40,42,39,43)

Polycyclic presentation:

2^9|a^2=i,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,d^b=di

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, 412412);