#456587

order	512
nilpotency class	3
isoclinism	isoclinic class of #456417
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 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)(6,11,16,15)(7,17)(8,21,23,25)(13,18,29,28)(19,34,33,22)(20,24)(27,35)(30,36);(5,6,12,11,10,16,9,15)(7,18,24,13,17,28,20,29)(8,14,21,26,23,32,25,31)(19,30,22,27,33,36,34,35);(5,7,10,17)(6,13,16,29)(8,22,23,34)(9,24,12,20)(11,18,15,28)(14,30,32,36)(19,25,33,21)(26,35,31,27);(2,4)(5,8)(6,14)(7,19)(9,25)(10,23)(11,26)(12,21)(13,27)(15,31)(16,32)(17,33)(18,30)(20,34)(22,24)(28,36)(29,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 363783);