#357461

order	512
nilpotency class	3
isoclinism	isoclinic class of #184767
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#537	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4694 conjugacy classes of subgroups; only the 650 normal subgroups are shown.

normal subgroups only

This group has 4694 conjugacy classes of subgroups — too many to draw. Only its 650 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,h^b=hi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264657);