#3553709

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2890 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,20)(21,23)(22,24)(25,26,29,30)(27,28,31,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,21,24,20)(26,30)(28,32);(3,7)(6,8)(10,13)(14,15)(17,20,22,21)(18,19,24,23)(25,27,29,31)(26,28,30,32);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,16)(13,15)(17,21,22,20)(18,23,24,19)(25,28,29,32)(26,27,30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326954, #72326989, #72327034
maximal normal subgroup of	#72326954, #72326989, #72327034
maximal quotient of	#72329040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3460905);