#3501363

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2582 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)(3,4)(5,7)(6,8)(10,13)(11,15)(17,18)(19,20)(21,24)(22,23)(26,30)(27,31);(1,3,6,7)(2,5,8,4)(11,15)(14,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,28)(29,30)(31,32);(1,4,6,5)(2,3,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,22)(18,23,24,19)(25,27,29,31)(26,28,30,32);(1,5,6,4)(2,7,8,3)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24)(25,28,29,32)(26,27,30,31);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,23)(19,24)(21,22)(25,27)(26,28)(29,31)(30,32)

Polycyclic presentation:

2^9|b^2=fgi,c^2=f,d^2=ghi,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	#72326682, #72326705, #72326749
maximal normal subgroup of	#72326682, #72326705, #72326749

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3408559);