#517634

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	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3498 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=i,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, 424830);