#6628982

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|b^2=f,e^2=i,b^a=bf,c^a=cg,d^a=dh,h^c=hi,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72344090, #72344091, #72344102
maximal subgroup of	D₄×#3025, #72252467, #72252468, …
maximal normal subgroup of	D₄×#3025, #72252467, #72252468, …
maximal quotient of	#72257970, C₂×#6628982, #72264449, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536178);