#4310310

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 2097152
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	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3380 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326664, #72326697, #72326724
maximal normal subgroup of	#72326664, #72326697, #72326724
maximal quotient of	#72328348

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4217506);