#6371669

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278865);