#4057260

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2894 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326798, #72326830, #72326872
maximal normal subgroup of	#72326798, #72326830, #72326872
maximal quotient of	#72328442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3964456);