#3898144

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2123 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=h,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,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, 3805340);