#111925

order	512
nilpotency class	3
isoclinism	isoclinic class of #111915
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#2657	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1570 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228198, #72235370, #72235390
maximal normal subgroup of	#72228198, #72235370, #72235390
maximal quotient of	#72228190

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19121);