#454989

order	512
nilpotency class	3
isoclinism	isoclinic class of #454986
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

1482 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)(7,12,18,27)(8,13)(9,16)(10,15)(11,14)(17,33,34,25)(19,26,24,28)(20,29,35,36)(21,32)(22,30)(23,31);(5,7)(6,12)(8,20)(9,18)(10,24)(11,19)(13,29)(14,26)(15,28)(16,27)(17,22)(21,35)(23,34)(25,31)(30,33)(32,36);(2,4)(5,8,9,21)(6,13,16,32)(7,17,18,34)(10,23,11,22)(12,25,27,33)(14,30,15,31)(19,35,24,20)(26,29,28,36);(6,14,16,15)(8,21)(12,26,27,28)(13,30,32,31)(17,34)(20,35)(22,23)(25,29,33,36)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ei,g^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,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=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, 362185);