#2109123

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

3282 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2016319);