#1409232

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1316428);