#114348

order	512
nilpotency class	3
isoclinism	isoclinic class of #114344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1136 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214129, #72214857, #72214883
maximal normal subgroup of	#72214129, #72214857, #72214883

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21544);