#115016

order	512
nilpotency class	3
isoclinism	isoclinic class of #115010
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22212);