#353181

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²	C₂×C₄³
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2412 conjugacy classes of subgroups; only the 842 normal subgroups are shown.

normal subgroups only

This group has 2412 conjugacy classes of subgroups — too many to draw. Only its 842 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260377);