#4430413

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2252 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4337609);