#3368559

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3559 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)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,16,15)(17,18,21,23)(19,20,24,22)(25,26,29,30)(27,28,31,32);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,23,20)(26,30)(27,31);(3,7)(5,8)(10,13)(14,15)(18,23)(19,24)(25,27)(26,28)(29,31)(30,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28)(29,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,31)(27,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,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	#72328620, #72328632, #72328711
maximal normal subgroup of	#72328620, #72328632, #72328711
maximal quotient of	#72330833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3275755);