#4306522

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 6291456
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2302 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=f,d^2=f,e^2=fgh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4213718);