#4028231

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1946 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)(3,7)(4,5)(6,8)(10,13)(11,15)(17,18,22,20)(19,23,21,24)(25,26,29,30)(27,31,32,28);(2,5)(6,7)(9,10,12,13)(11,16,15,14)(17,19,22,21)(18,23,20,24)(25,27,29,32)(26,28,30,31);(2,5)(6,7)(10,13)(14,16)(17,19,22,21)(18,24,20,23)(25,26,29,30)(27,31,32,28);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,23,21,24)(25,28)(26,27)(29,31)(30,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,21,22,19)(18,23,20,24)(25,26)(27,31)(28,32)(29,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=ghi,d^2=fghi,e^2=fgh,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, 3935427);