#128137

order	512
nilpotency class	4
isoclinism	isoclinic class of #128057
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

612 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,c^b=ch,d^b=dh,e^b=egi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308391, #72308394, #72308422
maximal normal subgroup of	#72308391, #72308394, #72308422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35333);