#455256

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362452);