#545525

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(5,6,8,7)(10,12)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(2,4)(9,10,13,12)(11,14,15,16)(18,22)(19,26)(24,28)(27,32);(1,2,3,4)(5,7,8,6)(11,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(9,11,13,15)(10,14,12,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(9,12,13,10)(11,16,15,14)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 452721);