#456671

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1388 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,h^c=hi,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, 363867);