#531002

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,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, 438198);