#348130

order	512
nilpotency class	3
isoclinism	isoclinic class of #348128
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1092 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255326);