#4430412

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	C₂×#2635	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2578 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)(7,12)(8,10)(9,11)(13,14)(15,16)(17,20)(18,19)(23,27)(26,28);(5,7,9,8)(6,10,11,12)(13,14,17,20)(15,16,19,18)(22,25)(26,28);(1,2,4,3)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,8,9,7)(6,10,11,12)(13,15)(14,16)(17,19)(18,20);(5,6)(7,10)(8,12)(9,11)(13,16)(14,19)(15,20)(17,18)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|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, 4337608);