#492908

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2443 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=be,e^a=eh,g^a=gi,h^a=hi,f^b=fhi,d^c=di,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225792, #72225797, #72227830, …
maximal normal subgroup of	#72225792, #72225797, #72227830, …
maximal quotient of	#72225785, #72268016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400104);