#455684

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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:

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

Polycyclic presentation:

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

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, 362880);