#4422948

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
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

2370 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=fg,d^2=fg,e^2=fh,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, 4330144);