#2355477

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3159 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=fghi,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	#72329131, #72329146, #72329223
maximal normal subgroup of	#72329131, #72329146, #72329223
maximal quotient of	#72330829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2262673);