#354031

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4106 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 4106 conjugacy classes of subgroups — too many to draw. Only its 738 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(2,4)(9,10)(11,17,19,14)(12,15)(13,16)(18,21,24,23)(20,22);(5,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,2,3,4)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(10,16)(14,17)(15,22)(21,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,b^a=be,c^a=cf,d^a=di,e^a=ei,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261227);