#1499123

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3082 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(18,21)(19,20)(26,28)(27,31);(2,5)(6,7)(10,14)(15,16)(17,18)(19,24)(20,23)(21,22);(9,11,13,12)(10,15,14,16)(17,19,22,20)(18,23,21,24)(25,26,29,28)(27,30,31,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,14,16)(17,20,22,19)(18,23,21,24)(25,27)(26,30)(28,32)(29,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21,22,18)(19,24,20,23)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|c^2=i,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336311, #72336360, #72336429
maximal normal subgroup of	#72336311, #72336360, #72336429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1406319);