#119906

order	512
nilpotency class	3
isoclinism	isoclinic class of #119905
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1048 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,g^b=gh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27102);