#1418352

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2423 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)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(19,23)(22,24)(25,26,29,32)(27,31,28,30);(1,3)(2,7)(4,5)(6,8)(9,11,13,16)(10,14,15,12)(25,26)(27,30)(28,31)(29,32);(2,4)(3,8)(11,16)(12,14)(17,18,21,20)(19,22,23,24)(25,27,29,28)(26,30,32,31);(1,4,6,2)(3,5,8,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32);(1,5)(2,8)(3,4)(6,7)(9,10)(11,14)(12,16)(13,15)(17,20,21,18)(19,24,23,22)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fghi,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,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, 1325548);