#2080888

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3223 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)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,28)(21,24)(22,25)(23,31)(27,30)(29,32);(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,24)(18,21,25,26)(20,29,31,27)(23,32,28,30);(5,6,7,8)(10,14)(13,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,31)(29,32);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,20,22,31)(18,23,25,28)(19,27,24,29)(21,30,26,32);(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,25)(20,30)(22,26)(23,29)(27,28)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=h,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, 1988084);