#115184

order	512
nilpotency class	3
isoclinism	isoclinic class of #115179
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1810 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1810 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 3 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22380);