#530384

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

2986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 437580);