#557399

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

2194 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)(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,16,20,19)(21,22,25,26)(23,24,27,28);(6,9)(7,11)(14,18)(16,19)(21,23,25,27)(22,24,26,28);(1,3,4,2)(13,15)(14,19)(16,18)(17,20)(21,24)(22,23)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,20)(21,23)(22,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,19)(14,15,18,20)(21,22)(23,24)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,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, 464595);