#2441491

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2926 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)(11,15)(12,13)(14,16)(17,18)(19,20)(21,23)(22,24)(26,27)(28,32);(2,3)(6,7)(9,10,13,12)(11,14,16,15)(17,19,21,24)(18,22,23,20)(26,27)(30,31);(4,8)(6,7)(10,12)(14,15)(19,24)(20,22)(25,26,29,27)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,23)(22,24)(25,27,29,26)(28,31,32,30);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|b^2=fgi,c^2=fghi,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329211, #72329218, #72329234
maximal normal subgroup of	#72329211, #72329218, #72329234
maximal quotient of	#72331595

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2348687);