#457582

order	512
nilpotency class	3
isoclinism	isoclinic class of #454676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1540 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi,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, 364778);