#479732

order	512
nilpotency class	3
isoclinism	isoclinic class of #478457
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

791 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2)(3,4)(5,6)(7,13,15,9)(8,16)(10,12)(11,19)(14,18,17,20);(5,7)(6,9)(8,17,19,14)(10,15)(11,18,16,20)(12,13);(5,8)(6,11,12,16)(7,14)(9,18,13,20)(10,19)(15,17);(2,4)(5,9)(6,7,12,15)(8,18,19,20)(10,13)(11,17)(14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386928);