#512374

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2213	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

830 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fi,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419570);