#2199596

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2237 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,23,24,21)(26,29)(27,28);(1,3,6,5)(2,7,4,8)(9,10,13,12)(11,14,16,15)(19,24)(21,23)(25,26,30,29)(27,32,28,31);(3,5)(7,8)(10,12)(14,15)(17,19)(18,21)(20,24)(22,23)(25,27,30,28)(26,31,29,32);(1,4,6,2)(3,7,5,8)(9,11,13,16)(10,14,12,15)(17,19)(18,21)(20,24)(22,23)(25,28,30,27)(26,31,29,32);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,29,30,26)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fgi,d^2=f,e^2=f,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, 2106792);