#4306682

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6,9,11)(7,8,12,10)(14,17)(18,19)(21,22,25,24)(23,27,28,26);(2,4)(5,7,9,12)(6,10,11,8)(14,17)(18,19)(22,24)(23,28);(1,2)(3,4)(13,14,16,17)(15,19,20,18)(21,23,25,28)(22,26,24,27);(5,8)(6,7)(9,10)(11,12);(5,7)(6,8)(9,12)(10,11)(13,15,16,20)(14,18,17,19)(21,24,25,22)(23,26,28,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,e^2=g,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, 4213878);