#232149

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,e^a=eh,c^b=cefi,d^b=dei,f^b=fh,g^b=gh,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, 139345);