#454988

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 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)(7,17,19,12)(8,13)(9,16)(10,14)(11,15)(18,26,34,33)(20,28,25,27)(21,36,35,29)(22,32)(23,30)(24,31);(5,7)(6,12)(8,21)(9,25)(10,20)(11,19)(13,29)(14,27)(15,17)(16,28)(18,22)(23,34)(24,35)(26,30)(31,36)(32,33);(2,4)(5,8,11,24)(6,13,15,31)(7,18,19,34)(9,23,10,22)(12,26,17,33)(14,32,16,30)(20,35,25,21)(27,29,28,36);(6,14,15,16)(12,27,17,28)(13,30,31,32)(26,36,33,29)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,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, 362184);