#1605371

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,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^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300845, #72300875, #72300882
maximal normal subgroup of	#72300845, #72300875, #72300882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1512567);