#518716

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3306 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=fi,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425912);