#355268

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262464);