#146650

order	512
nilpotency class	4
isoclinism	isoclinic class of #12778
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10745
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,6,12,8,13,7,23)(3,14,18,30,19,31,17,25)(4,11,20,29,5,21,22,10)(9,24,27,15,28,32,26,16);(1,3,7,18,8,19,6,17)(2,9,12,27,13,28,23,26)(4,15,20,24,5,16,22,32)(10,25,21,31,29,30,11,14);(1,4)(2,10)(3,15)(5,8)(6,22)(7,20)(9,25)(11,23)(12,21)(13,29)(14,26)(16,19)(17,32)(18,24)(27,31)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279536, #72317333, #72317388, …
maximal normal subgroup of	#72279536, #72317333, #72317388, …
maximal quotient of	#72279868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53846);