#494738

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#35513	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232771, #72232800, #72232831, …
maximal normal subgroup of	#72232771, #72232800, #72232831, …
maximal quotient of	#72232766

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401934);