#651023

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264052, D₄×#2407, #72264105, …
maximal normal subgroup of	#72264052, D₄×#2407, #72264105, …
maximal quotient of	#72264557, #72264562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 558219);