#4399217

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2548 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)(6,9)(10,12)(13,14,19,20)(15,17,18,16)(22,25)(23,27);(13,15,19,18)(14,16,20,17)(22,25)(26,28);(1,2)(3,4)(13,16,19,17)(14,15,20,18)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,17,19,16)(14,15,20,18);(5,7,8,11)(6,10,9,12)(13,18,19,15)(14,16,20,17)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=f,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4306413);