#6373480

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280676);