#393779

order	512
nilpotency class	3
isoclinism	isoclinic class of #195963
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

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	C₂×#2656	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2424 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 2424 conjugacy classes of subgroups — too many to draw. Only its 426 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,h^b=hi,d^c=di

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, 300975);