#146574

order	512
nilpotency class	4
isoclinism	isoclinic class of #11549
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1802 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)(3,4)(5,6)(7,8)(9,10,14,19,16,11,21,23)(12,17,13,22,15,20,24,18);(2,4)(9,11,15,17,16,10,12,20)(13,18,21,19,24,22,14,23);(6,8)(9,12,16,15)(10,17,11,20)(13,21,24,14)(18,19,22,23)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,d^b=dh,f^b=fhi,h^b=hi,f^d=fi,g^d=gi,g^f=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268016, #72268020, #72268021
maximal normal subgroup of	#72268016, #72268020, #72268021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53770);