#454066

order	512
nilpotency class	3
isoclinism	isoclinic class of #454021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2205 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)(7,19)(8,24)(9,26)(10,16)(11,18)(12,17)(13,22)(14,25)(15,27)(20,33)(21,36)(23,30)(28,32)(29,35)(31,34);(2,4)(5,7)(6,13)(8,20)(9,21)(10,22)(11,28)(12,23)(14,31)(15,29)(16,19)(17,32)(18,30)(24,36)(25,35)(26,33)(27,34);(1,2)(3,4)(5,8)(6,14)(7,20)(9,10)(11,27)(12,25)(13,29)(15,16)(17,26)(18,24)(19,31)(21,22)(23,35)(28,34)(30,33)(32,36);(2,4)(5,9,10,8)(6,15,16,14)(7,21,22,20)(11,27,12,25)(13,29,19,31)(17,26,18,24)(23,35,28,34)(30,36,32,33)

Polycyclic presentation:

2^9|d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,f^c=fi,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, 361262);