#4306787

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,29)(30,32);(9,11)(10,12)(13,15)(14,16)(17,18,20,22)(19,21,24,23)(26,29)(30,31);(2,5)(6,7)(10,14)(11,15)(17,18,20,22)(19,21,24,23)(25,27,28,32)(26,30,29,31);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,29)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^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.

Relationships

maximal subgroup of	#72303671, #72303677, #72303689
maximal normal subgroup of	#72303671, #72303677, #72303689

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4213983);