#455566

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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

1160 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)(5,6,10,15)(7,16,17,11)(8,13,21,28)(9,12,22,14)(18,29,30,24)(19,23,31,25)(20,26,33,27)(32,34,36,35);(5,7,9,19,10,17,22,31)(6,11,14,23,15,16,12,25)(8,18,20,32,21,30,33,36)(13,24,27,34,28,29,26,35);(6,12)(7,17)(9,22)(11,23)(13,26)(14,15)(16,25)(20,33)(24,35)(27,28)(29,34)(32,36);(2,4)(5,8)(6,13)(7,18)(9,20)(10,21)(11,24)(12,26)(14,27)(15,28)(16,29)(17,30)(19,32)(22,33)(23,34)(25,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,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, 362762);