#537422

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2314 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)(7,8)(11,12)(14,18)(16,19);(5,6,10,9)(7,11,8,12)(13,14,17,18)(15,16,20,19)(22,25)(26,28);(1,3,4,2)(5,7,10,8)(6,11,9,12)(13,15,17,20)(14,19,18,16)(21,22)(23,26)(24,25)(27,28);(5,8,10,7)(6,12,9,11)(13,15)(14,16)(17,20)(18,19);(5,9,10,6)(7,12,8,11)(13,16)(14,15)(17,19)(18,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fgi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=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, 444618);