#6375879

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ 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)(6,9)(7,11)(13,14)(15,23)(16,27)(17,19)(18,31)(20,26)(21,28)(22,30)(24,36)(25,39)(29,35)(32,33)(34,44)(37,38)(40,41)(42,43);(1,3,4,2)(5,6,8,9)(7,10,11,12)(14,19)(15,24)(21,27)(22,31)(25,38)(26,32)(35,42)(40,44);(6,9)(10,12)(13,15,17,24)(14,20,19,33)(16,25,28,38)(18,32,30,26)(21,34,27,41)(22,36,31,23)(29,44,43,40)(35,39,42,37);(5,7,8,11)(6,10,9,12)(13,16,17,28)(14,21,19,27)(15,25,24,38)(18,29,30,43)(20,34,33,41)(22,35,31,42)(23,37,36,39)(26,40,32,44);(16,28)(21,27)(25,38)(29,43)(34,41)(35,42)(37,39)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,g^c=gi,e^d=ei

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, 6283075);