#554126

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

2024 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)(5,6)(7,11)(8,9)(10,12)(13,14,17,16)(15,19,20,18)(21,22)(23,24)(25,26)(27,28);(6,9)(10,11)(14,16)(18,19)(21,23,26,28)(22,27,25,24);(1,3,4,2)(13,14,17,16)(15,18,20,19)(21,24,26,27)(22,28,25,23);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,22)(23,27,28,24);(5,6)(7,10)(8,9)(11,12)(13,16,17,14)(15,19,20,18)(21,23)(22,24)(25,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 461322);