#416571

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35001	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2968 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 2968 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei

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, 323767);