#4306972

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 12582912
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

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

8172 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)(3,4)(5,6,8,7)(9,10,12,11)(15,19)(17,20);(6,7)(10,11)(13,14,16,18)(15,17,19,20);(2,4)(6,7)(9,10,12,11);(13,15)(14,17)(16,19)(18,20);(1,2)(3,4)(5,7,8,6)(9,11,12,10)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4214168);