#517755

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3230 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)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(6,8)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(1,3,4,2)(13,15,17,22)(14,18,20,26)(16,23,24,21)(19,27,28,25);(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,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, 424951);