#4306809

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=f,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.

Relationships

maximal subgroup of	#72303662, #72303669, #72303671
maximal normal subgroup of	#72303662, #72303669, #72303671

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4214005);