#512306

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

639 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|b^2=efi,d^2=eh,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^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,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, 419502);