#2199595

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1974 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 2106791);