#6371352

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4491 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)(3,4)(5,6,11,15,12,8,10,16)(7,17,20,13,22,28,21,18)(9,14,26,29,24,23,25,30)(19,31,34,27,33,36,35,32);(2,4)(5,7,11,20,12,22,10,21)(6,13,15,28,8,18,16,17)(9,19,26,34,24,33,25,35)(14,27,29,36,23,32,30,31);(5,8)(6,12)(7,18)(9,23)(10,15)(11,16)(13,22)(14,24)(17,20)(19,32)(21,28)(25,29)(26,30)(27,33)(31,34)(35,36);(5,9)(6,14)(7,19)(8,23)(10,25)(11,26)(12,24)(13,27)(15,29)(16,30)(17,31)(18,32)(20,34)(21,35)(22,33)(28,36);(9,24)(14,23)(19,33)(25,26)(27,32)(29,30)(31,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278548);