#564628

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2356 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)(6,9)(7,11)(13,14,17,20)(15,16,19,18)(22,26)(24,27);(5,6,8,9)(7,12,11,10)(13,15,17,19)(14,18,20,16)(22,26)(23,28);(5,6,8,9)(7,10,11,12)(13,16,17,18)(14,19,20,15)(21,22)(23,27)(24,28)(25,26);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23,25,28)(22,27,26,24);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24,25,27)(22,28,26,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 471824);