#495152

order	512
nilpotency class	4
isoclinism	isoclinic class of #480454
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fi,d^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402348);