#427827

order	512
nilpotency class	3
isoclinism	isoclinic class of #419895
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335023);