#455818

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	#23387	C₂
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)(7,16)(8,13)(9,22)(10,15)(11,18)(12,20)(14,24)(17,31)(19,27)(21,29)(23,28)(25,36)(26,33)(30,34)(32,35);(5,7,9,19,10,18,24,35)(6,11,14,27,15,16,22,32)(8,17,20,34,21,33,23,36)(12,25,13,26,28,30,29,31);(5,8)(6,12)(7,17,18,33)(9,23)(10,21)(11,25,16,30)(13,22)(14,29)(15,28)(19,36,35,34)(20,24)(26,27,31,32);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,26)(12,22)(14,28)(15,29)(16,31)(18,33)(19,34)(23,24)(25,32)(27,30)(35,36)

Polycyclic presentation:

2^9|b^2=f,c^2=e,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^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, 363014);