#507174

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

612 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)(13,14,20,27)(15,29,33,44)(16,34,36,24)(17,23,37,43)(18,38,31,26)(19,39,30,25)(21,40,35,28)(22,41,42,32);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,30,36,21,17,31,41,20,33,19,16,35,37,18,32)(14,22,38,43,28,34,39,44,27,42,26,23,40,24,25,29);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,33,35,41,19,17,20,36,31,15,21,32,30,37)(14,23,25,42,40,44,26,34,27,43,39,22,28,29,38,24);(6,11)(10,12)(13,17,19,32,21,33,18,36,20,37,30,41,35,15,31,16)(14,24,26,44,28,22,25,23,27,34,38,29,40,42,39,43)

Polycyclic presentation:

2^9|a^2=i,b^2=eghi,c^2=e,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^c=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, 414370);