#6375870

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 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)(15,19)(16,26)(17,29)(18,21)(20,27)(22,31)(23,37)(24,34)(25,41)(28,36)(30,35)(32,39)(33,40)(38,44)(42,43);(1,3,4,2)(5,6,8,9)(7,10,11,12)(13,15,18,24)(14,19,21,34)(16,23,28,38)(17,25,31,40)(20,32,35,43)(22,33,29,41)(26,37,36,44)(27,39,30,42);(6,9)(10,12)(13,16,18,28)(14,20,21,35)(15,23,24,38)(17,30,31,27)(19,32,34,43)(22,36,29,26)(25,42,40,39)(33,44,41,37);(5,7,8,11)(6,10,9,12);(14,21)(15,24)(20,35)(22,29)(23,38)(25,40)(26,36)(39,42)

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,e^a=ei,g^a=gi,c^b=cf,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283066);